A survey on the existence of G-Designs

A Survey on the Existence of G-Designs

Peter Adams, Darryn Bryant, Melinda BuchananDepartment of Mathematics, University of Queensland, Qld 4072, Australia,E-mail: [email protected]; [email protected]; [email protected]

Received April 5, 2007; revised July 17, 2007

Published online 2 November 2007 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/jcd.20170

Abstract: A G-design of order n is a decomposition of the complete graph on n vertices intoedge-disjoint subgraphs isomorphic to G. We survey the current state of knowledge on theexistence problem for G-designs. This includes references to all the necessary designs andconstructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16:373–410, 2008

Keywords: G-design; graph decomposition

1. INTRODUCTION

In this article, all graphs are simple. Let G be a graph. A G-decomposition of a graph K isa set of subgraphs of K, each isomorphic to G, whose edge sets partition the edge set of K.A G-design of order n is a G-decomposition of the complete graph on n vertices, which isdenoted by Kn. The spectrum for a graph G is the set S of positive integers given by n ∈ S

if and only if there exists a G-design of order n.Numerous articles have been written on the existence of G-designs, including several

surveys [30,39,75,113] and a book [34]. The spectrum problem has been considered for alarge number of graphs and families of graphs. There has also been a lot of work done ofvariations and generalizations of G-designs, and on G-designs with additional properties.However, here we restrict ourselves to the spectrum problem for G-designs as definedabove. The objective is to give a comprehensive statement of the current state of knowledge,including references to articles which give all the necessary proofs or constructions. Thearticle serves as verification for Theorem 24.45 in [39], and also contains a few resultswhich have been established since [39] was written, and a few which were overlooked.Some of the new results are from recently published articles and some are new G-designsand families of G-designs constructed here. We also mention a few anomalies and errors inpublished results.

The article is divided into several sections. Complete graphs and trees are covered inSections 2 and 3, respectively. The spectrum problem is completely solved for cycles,

Journal of Combinatorial Designs© 2007 Wiley Periodicals, Inc.

373

374 ADAMS, BRYANT AND BUCHANAN

matchings, paths, and stars, and Section 4 contains four corresponding subsections in whichthese results are given. Section 5 covers the remaining families of graphs for which thespectrum problem has been considered. It is divided into seven subsections on cubes, graphsof geometric solids, complete bipartite graphs, even graphs, theta graphs, unions of graphs,and miscellaneous graphs. The spectrum problem has also been considered for all graphswith up to five vertices, and for all graphs with six vertices and up to eight edges. Theseresults are covered in Section 6. An appendix containing several new G-designs is includedat the end of the article.

For any graph G, there are three obvious necessary conditions for the existence of aG-design of order n:

(1) |V (G)| ≤ n for n > 1;(2) n(n − 1) ≡ 0 (mod 2|E(G)|); and(3) n − 1 ≡ 0 (mod d) where d is the greatest common divisor of the degrees of the

vertices in G.

Wilson [116] has shown that these conditions are asymptotically sufficient. That is, forany given graph G, there exists an integer N(G) (N(G) is exponentially large) such that ifn > N(G) then Conditions (2) and (3) above are sufficient for the existence of a G-designof order n.

The spectrum for a graph G′ with isolated vertices is the same as that for the graphG obtained from G′ by removing its isolated vertices, except precisely when there is aG-design of order n > 1 and |V (G′)| > n. Thus, throughout the article we consider onlygraphs having no isolated vertices.

2. COMPLETE GRAPHS

When G is a complete graph on k vertices, a G-design of order n is better known as a (v, k, 1)-BIBD with v = n. Table I summarizes the current state of knowledge on the spectrumproblem for Kk with k ≤ 9. This table is in agreement with Table 3.3 in the recent surveyby Abel and Greig [7]. The spectrum problem for K2 is trivial and the spectrum problemfor K3 was solved in 1847 by Kirkman [87]. A K3-design is a Steiner triple system. Thespectrum problem for K4 and K5 was solved by Hanani [72] in 1961.

Substantial progress has been made on the spectrum problem for K6 and numerousarticles have appeared in the literature. In particular, Mills [99] compiled a summary ofthe state of knowledge in 1996 which includes the necessary references to constructionsand designs from other sources. At that point, there were 55 unresolved values inthe spectrum for K6. The 1996 survey [5] by Abel and Greig lists 53 unresolvedcases. The list of unresolved cases has since been reduced to the 29 values given inTable I. The 26 new results, not covered by Mills’ survey [99], are as follows. Thenon-existence of K6-design of order 46 is established by Houghten et al. in [78]. Abeland Greig [6] have constructed a K6-design of order n for each of the 14 values of nin {276, 466, 706, 741, 946, 1096, 1246, 1396, 1456, 1486, 1521, 1611, 1671, 2031}, andAbel et al. [3] have constructed a K6-design of order n for each of the 11 values ofn in {141, 196, 201, 291, 496, 526, 766, 916, 1221, 1251, 1851}. The article [3] includesalternative constructions for five designs which were given incorrectly in an earlier article ofAbel and Mills [8]. We note that the non-existence of K6-designs of order n for n ∈ {16, 21}Journal of Combinatorial Designs DOI 10.1002/jcd

EXISTENCE OF G-DESIGNS 375

TABLE I. The Spectrum for Complete Graphs with up to 9 vertices

k Spectrum for Kk Possible exceptions

2 all n ∅3 n ≡ 1 or 3 (mod 6) ∅4 n ≡ 1 or 4 (mod 12) ∅5 n ≡ 1 or 5 (mod 20) ∅6 n ≡ 1 or 6 (mod 15) n ∈ {51, 61, 81, 166, 226, 231, 256, 261, 286, 316, 321,

and n �∈ {16, 21, 36, 46} 346, 351, 376, 406, 411, 436, 441, 471, 501, 561,591, 616, 646, 651, 676, 771, 796, 801}

7 n ≡ 1 or 7 (mod 42) n = 42t + 1 for t ∈ {2, 3, 5, 6, 12, 14, 17, 19, 22, 27,and n �= 43 33, 37, 39, 42, 47, 59, 62};

and n = 42t + 7 for t ∈ {3, 19, 34, 39}8 n ≡ 1 or 8 (mod 56) n = 56t + 1 for t ∈ {2, 3, 4, 5, 6, 7, 14, 19, 20, 21, 22,

24, 25, 26, 27, 28, 31, 32, 34, 35, 39, 40, 46, 52, 59, 61,62, 67}; andn = 56t + 8 for t ∈ {3, 11, 13, 20, 22, 23, 25, 26, 27, 28}

9 n ≡ 1 or 9 (mod 72) n = 72t + 1 for t ∈ {2, 3, 4, 5, 7, 11, 12, 15, 20, 21, 22,24, 27, 31, 32, 34, 37, 38, 40, 42, 43, 45, 47, 50, 52, 53,56, 60, 61, 62, 67, 68, 75, 76, 84, 92, 94, 96, 102, 132,174, 191, 194, 196, 201, 204, 209};and n = 72t + 9 for t ∈ {2, 3, 4, 5, 12, 13, 14, 18, 22, 23,25, 26, 27, 28, 31, 33, 34, 38, 40, 41, 43, 46, 47, 52, 59,61, 62, 67, 68, 76, 85, 93, 94, 102, 103, 139, 148, 174,183, 192, 202, 203, 209, 229}

follows from Theorem 2.2 below, and the non-existence of a K6-design of order n = 36follows from Theorem 2.3 below.

Abel [1] and Greig [69] establish the results given in Table I for k ∈ {7, 8, 9}. Thesetwo articles contain several new designs which were listed as undecided in [5]. Greig [69]cites results of Abel [2], Buratti and Zuanni [44], and Janko and Tonchev [85] to establishresults. The non-existence of a K7-design of order 43 follows from Theorem 2.3 below.

For k ≥ 10, the known Kk-designs are given by Theorem 2.1 (and Wilson’sTheorem [116]).

Theorem 2.1. Let q be a prime power and let n, r, and s be integers such that n ≥ 2 and2 ≤ r < s.

� There exists a Kq-design of order qn (affine geometries) (see Wallis [114]).� There exists a Kq+1-design of order qn + · · · + q + 1 (projective geometries) (see

Wallis [114]).� There exists a Kq+1-design of order q3 + 1 (unital designs) (see Assmus and Key [24]).� There exists a K2r−1-design of order 2r−1(2r − 1) (oval designs) Bose and Shrikhande

[35].� There exists a K2r -design of order 2r+s + 2r − 2s (Denniston designs) Denniston [55].

Obvious necessary conditions for the existence of a Kk-design of order n are

� n = 1 or n ≥ k;

Journal of Combinatorial Designs DOI 10.1002/jcd


TABLE II. The Spectrum for Trees With up to 8 Edges

m Spectrum for trees with m edges Exceptions (T1, . . . , T10 as in Fig. 1)

1 All n ∅2 n ≡ 0 or 1 (mod 4) ∅3 n ≡ 0 or 1 (mod 3), n �= 3 There is no K1,3-design of order 44 n ≡ 0 or 1 (mod 8) ∅5 n ≡ 0 or 1 (mod 5), n �= 5 There is no K1,5-design of order 6

There is no Ti-design of order 6 for i ∈ {1, 2}6 n ≡ 0, 1, 4 or 9 (mod 12), n �= 4 There is no K1,6-design of order 97 n ≡ 0 or 1 (mod 7), n �= 7 There is no K1,7-design of order 8

There is no Ti-design of order 8for i ∈ {3, 4, 5, 6, 7, 8, 9, 10}

8 n ≡ 0 or 1 (mod 16) ∅

� n(n − 1) ≡ 0 (mod k(k − 1)); and� n ≡ 1 (mod (k − 1)).

Further necessary conditions for the existence of a Kk-design of order n follow from Fisher’sInequality [62], the Bruck–Ryser–Chowla Theorem [37,48] and the fact that there exists aKk-design of order k2 if and only if there exists a Kk+1-design of order k2 + k + 1. Proofsof these results are given in Wallis’ text [114], for example. The implications of these resultsfor the existence of Kk-designs are given in the following two theorems.

Theorem 2.2 ([62]). There is no Kk-design of order n for any n in the range k < n <

k2 − k + 1.

Theorem 2.3 ([37,48]). If k ≡ 2 or 3 (mod 4) and k − 1 is not the sum of two integersquares then there is no Kk-design of order k2 − k + 1 and no Kk−1-design of order k2 −2k + 1.

Finally, we have the result of Lam et al. [92] that there is no projective plane of order 10.

Theorem 2.4 ([92]). There is no K10-design of order 100 and no K11-design of order111.

3. TREES

A tree is a connected graph which contains no cycles. The spectrum problem for treeswith at most nine vertices is completely solved by Huang and Rosa [83] and the resultsare summarized in Table II (the graphs T1, T2, . . . , T10 referenced in the table are shown inFig. 1). There are two infinite families of trees for which the spectrum problem is completelysolved, namely paths and stars, see Theorems 4.3 and 4.4. A star is a complete bipartitegraph with one part of size 1 and one part of size k ≥ 1, and is denoted by K1,k or Sk.

Numerous existence results for G-designs, in particular when G is a tree, have beenestablished under the guise of graph labelings. Graph labelings were introduced by Rosa[106] in 1967 to tackle Ringel’s Conjecture [104], which states that for any tree T withx + 1 vertices, there exists a T-design of order 2x + 1.



FIGURE 1. Trees referenced in Table II.

The two types of labelings which are of most relevance to the existence of G-designsare ρ-labelings and ρ+-labelings. The definitions of these and other graph labelings canbe found in [39]. Here, we are interested only in their implications for the existence ofG-designs, as given by the following two theorems. The first is due to Rosa [106] and thesecond to El Zanati et al. [61].

Theorem 3.1 ([106]). If a graph G has a ρ-labeling, then there exists a G-design of order2|E(G)| + 1.

Theorem 3.2 ([61]). If a graph G has a ρ+-labeling, then there exists a G-design of ordern for all n ≡ 1 (mod 2|E(G)|).

Most of the results presented in the remainder of this section (and some in other sections)are obtained from results on graph labelings by applying the above two theorems. Note thatvarious other types of labelings are special ρ- or ρ+-labelings, and thus also yield G-designsas described in the above two theorems. For example, α-labelings, introduced by Rosa in[106], are a special class of ρ+-labelings, and β-labelings (probably more well known asgraceful labelings) are a special class of ρ-labelings. See [39] for a description of graphlabelings, the relationships between them, and their implications for existence of G-designs.

Theorems 3.3 and 3.4 give the main results on G-designs arising from ρ-labelings andρ+-labelings, respectively, when G is a tree. The diameter of a graph is the maximum overall pairs {x, y} of vertices of the distance between x and y. A vertex of degree 1 in a graphis called an endvertex. The base of a graph G is the graph obtained from G by removing itsendvertices. A caterpillar is a tree whose base is a path, and a lobster is a tree whose baseis a caterpillar. A comet is a graph obtained from a star by replacing each edge with a pathof length k for some fixed k. A tree is symmetric if it can be rooted so that any two verticesin the same level have the same degree.

Theorem 3.3. Let T be a tree belonging to one of the following families.

� Trees with at most 55 vertices [36].� Trees with at most 4 endvertices [82].



� Trees of diameter at most 5 [79,121].� Symmetric trees [101].

Then there exists a T-design of order 2|E(T )| + 1.

Theorem 3.4. Let T be a tree belonging to one of the following families.

� Trees with at most 21 vertices [68].� Trees of diameter at most 5 [61].� Symmetric trees of diameter 4 [59].� Caterpillars [106].� Comets [61].

Then there exists a T-design of order n for all n ≡ 1 (mod 2|E(T )|).A few more sporadic results on labelings of trees, not encompassed by Theorems 3.3

and 3.4 can be found in Gallian’s survey [63]. Theorems 3.5 and 3.6 are proven in Huangand Rosa [83], and Theorem 3.7 is due to Dobson [57].

Theorem 3.5 ([83]). Let T be a caterpillar or lobster with m + 1 vertices. If n ≡0 or 1 (mod 2m), there exists a T-design of order n. Moreover, if m = 2α for some integerα ≥ 0, then n ≡ 0 or 1 (mod 2m) is also necessary for existence.

Theorem 3.6 ([83]). Let T be a tree with m + 1 vertices. If T contains a vertex of degreed such that d ≥ 1

2 (m + 3), then there does not exist a T-design of order m + 1.

Theorem 3.7 ([57]). Let T be a tree with n + 1 vertices, let x be a vertex in T and supposeeither of the following holds.

� The graph obtained from T by removing x (and all the edges incident with x) has at

least n −√

n

4+2√

2isolated vertices.

� For a non-negative integer d, the diameter of T is at most d + 2, and the graph obtainedfrom T by removing x (and all the edges incident with x) has at least n − cn isolatedvertices where c = (

√1 + (4 + 4d)2 − 4 − 4d)2.

Then there exists a T-design of order 2n + 1.

4. CYCLES, MATCHINGS, PATHS, AND STARS

We group cycles, matchings, paths, and stars together in this section as the spectrum problemhas been completely settled for every graph in each of these families.

A. Cycles

The cycle with m vertices is denoted by Cm. The spectrum problem for Cm has a long historydating back to the solution of the spectrum problem for C3 by Kirkman [87] in 1847. AC3-design is also known as Steiner triple system (see Colbourn and Rosa’s text [50]). Thespectrum problem for Cm has now been solved for all m. The case m is odd was settledby Alspach and Gavlas [22] in 2001 and the case m is even was settled by Sajna [108] in



2002. The spectrum problem for Cm had been settled previously for numerous values of mincluding several infinite families of values and all m ≤ 50 [25]. Crucial ingredients in theeventual complete solution were Sotteau’s Theorem [109] and the result of Hoffman et al.[76]. These results reduced the problem to one of settling the small values of n for each m.Several surveys have been written on cycle decompositions [38,42,95]. In particular, [38]covers the history of the spectrum problem for Cm.

Theorem 4.1 ([22,108]). Let m ≥ 3. There exists a Cm-design of order n if and only if

� n = 1 or n ≥ m;� n is odd; and� n(n − 1) ≡ 0 (mod 2m).

B. Matchings

The graph consisting of k vertex disjoint edges is called a k-matching and is denoted by Mk.The following theorem, which completely settles the spectrum problem for Mk, is an easyconsequence of the results of Alon [20], Ellingham and Wormald [58], McDiarmid [98], orde Werra [115].

Theorem 4.2. Let k ≥ 1. There exists an Mk-design of order n if and only if

� n = 1 or n ≥ 2k; and� n(n − 1) ≡ 0 (mod 2k).

C. Paths

The path with m vertices is denoted by Pm. The spectrum problem for Pm was completelysettled for all m ≥ 1 by Tarsi [111] in 1983. Note that paths are trees (see Section 3).

Theorem 4.3 ([111]). Let m ≥ 2. There exists a Pm-design of order n if and only if

� n = 1 or n ≥ m; and� n(n − 1) ≡ 0 (mod 2m − 2).

D. Stars

The k-star, denoted by Sk, consists of a vertex x of degree k, its k neighbors, and the k edgesjoining x to its neighbors. A star is a tree (see Section 3) and is also a complete bipartitegraph (see Subsection 5.C). The spectrum problem for k-stars was completely solved forall k ≥ 1 in 1975 by Yamamoto et al. [117], and independently in 1979 by Tarsi [110].Theorem 4.4 gives the result.

Theorem 4.4 ([117]). Let k ≥ 1. There exists an Sk-design of order n if and only if

� n = 1 or n ≥ 2k; and� n(n − 1) ≡ 0 (mod 2k).



TABLE III. The Spectrum for Cubes

d Spectrum for the d-cube Possible exceptions

Even n ≡ 1 (mod d2d) ∅3 n ≡ 1 or 16 (mod 24) ∅5 n ≡ 1 or 96 (mod 160) ∅d ≥ 7 and odd n ≡ 1 or t (mod d2d) where t

is the unique integer satisfyingt ≡ 0 (mod 2d), t ≡ 1 (mod d)and 0 ≤ t < d2d

n ≡ t (mod d2d) with n notcovered by Theorems 5.1, 5.2,or Wilson’s Theorem [116]

5. OTHER FAMILIES

In this section, we give results on the spectrum problem for some further families of graphs,namely cubes, graphs of some geometric solids, complete bipartite graphs, even graphs,theta graphs, unions of graphs, and some other miscellaneous families.

A. Cubes

The d-cube or cube of dimension d, denoted by Qd , is the graph with vertex set consistingof all binary strings of length d and with two vertices adjacent if and only if they differ inexactly one coordinate.

The 1-cube is a single edge and the spectrum is trivially the set of all positive integers.The 2-cube is a 4-cycle and the spectrum is well known to be all n ≡ 1 (mod 8) (seeSubsection 4.A). The spectrum problem for the 3-cube was settled by Maheo in 1980 [97],and a solution is also given in [40]. In 1981, Kotzig [91] established necessary conditionsfor the existence of a d-cube design of order n and proved that there exists a Qd-design oforder n for all n ≡ 1 (mod d2d). This result completely settles the spectrum problem for thecase d is even. The spectrum for the 5-cube was settled in 2006 [41]. The spectrum problemfor the d-cube is currently unresolved for odd d ≥ 7. The above results on the spectrumproblem for the d-cube are summarized in Table III. Two further results, due to El Zanatiand Vanden Eynden [60] and Buratti [43], which prove the existence of sporadic familiesof d-cube designs, are given in Theorems 5.1 and 5.2.

Theorem 5.1 ([60]). Let d ≥ 1 be odd, let e ≥ d be such that 2e ≡ 1 (mod d) and let sbe the order of 2 (mod d). If t is a non-negative integer and n = 2e(t(2s − 1) + 1), thenthere exists a d-cube-design of order n.

Theorem 5.2 ([43]). Let d ≥ 1 be odd and let e ≥ d be such that 2e ≡ 1 (mod d). If t isa non-negative integer and n = 2e(d2dt + 1), then there exists a d-cube-design of order n.

B. Graphs of Geometric Solids

Results have been obtained on the spectrum problem for the graphs of each of the fivePlatonic solids and also for the cuboctahedron. These six graphs are shown in Figure 2 andthe results are summarized in Table IV.

The spectrum problems for the tetrahedron, the cube, the octahedron, and thecuboctahedron are completely solved. The tetrahedron, being the complete graph on four



FIGURE 2. Graphs of some geometric solids.

vertices, is covered in Section 2, the cube is covered in Subsection 5.A, the spectrumproblem for the octahedron was solved in 1988 by Horak and Rosa [77], and thespectrum problem for the cuboctahedron was completely solved by Grannell et al. [67].The spectrum problems for the dodecahedron and the icosahedron were partially settledin [12]. In [12], the authors also claim to have established the existence of various otherunspecified dodecahedron and icosahedron designs. These designs are given in ExamplesA.1–A.4 in the Appendix (and included in Table IV).

TABLE IV. The Spectrum for Graphs of Some Geometric Solids

Graph Spectrum Possible exceptions

Tetrahedron n ≡ 1 or 4 (mod 12) ∅Cube n ≡ 1 or 16 (mod 24) ∅Octahedron n ≡ 1 or 9 (mod 24) ∅

and n �= 9Dodecahedron n ≡ 1, 16, 25 or 40 (mod 60) n ≡ 16 (mod 60) with n ≥ 136

and n �= 16 n ≡ 25 (mod 60) with n ≥ 85n ≡ 40 (mod 60) with n ≥ 100

Icosahedron n ≡ 1, 16, 21 or 36 (mod 60) n ≡ 16 (mod 60) with n ≥ 76n ≡ 21 (mod 60) with n ≥ 21n ≡ 36 (mod 60) with n ≥ 36

Cuboctahedron n ≡ 1 or 33 (mod 48) ∅Designs covered by Wilson’s Theorem [116] are ignored in the listed possible exceptions for thedodecahedron and the icosahedron.



TABLE V. The Spectrum for Some Families of Complete Bipartite Graphs

Graph Spectrum

K1,k {1} ∪ {n : k divides(

n

2

)and n ≥ 2k}

K2α,2β for all α, β ≥ 1 n ≡ 1 (mod 2α+β+1)K2,3 n ≡ 0, 1, 4 or 9 (mod 12) and n /∈ {4, 9, 12}K3,3 n ≡ 1 (mod 9) and n �= 10

C. Complete Bipartite Graphs

The complete bipartite graph with one part of size s and one part of size t is denoted byKs,t . The spectrum problem for complete bipartite graphs has been completely settled inseveral cases, see Table V. In the case s = 1, we have stars (see Section 4). In 1966, Rosa[106] showed that every complete bipartite graph has an α-labeling, and thus establishedTheorem 5.3 (see Theorem 3.2). This result completely settles the spectrum problem incases where s and t are both powers of 2. In addition, Bermond et al. [28] and Guy andBeineke [70] settle the spectrum problem for K2,3 and K3,3, respectively.

Theorem 5.3 ([106]). There exists a Ks,t-design of order n for all n ≡ 1 (mod 2st).

Some non-existence results for Ks,t-designs have also been proven and are given in thefollowing theorem. These results are due to de Caen and Hoffman [45] and Graham andPollak [66].

Theorem 5.4 ([45,66]). Let s and t be any positive integers. There is no Ks,t-design oforder n for 1 < n < 2st. Moreover, for s, t ≥ 2 there does not exist a Ks,t-design of order 2st.

D. Even Graphs

In this subsection, we discuss the spectrum problem for even graphs. A graph is even if eachvertex has even degree. The spectrum problem for even graphs with 10 or fewer edges iscompletely settled. Table VI summarizes the results.

A 2-regular graph is a union of vertex disjoint cycles and we denote by Cm1 ∪ Cm2 ∪ · · · ∪Cmt the 2-regular graph consisting of t vertex disjoint cycles of lengths m1, m2, . . . , mt .An even graph which is connected is called a closed trail, and a closed trail with m edgesis a called a closed m-trail. We split the discussion of the results for even graphs into threeparts: 2-regular graphs, closed trails, and other even graphs.

The spectrum problem for each 2-regular graph with at most ten vertices is completelysolved in [15] and Table VI includes the results. Several instances had already been settledin earlier articles. For example, the spectrum for C3 ∪ C3 was settled by Horak and Rosa[77]. It is shown by Kohler [89] that there is no (C4 ∪ C5)-design of order 9 and it isalso well known (though at present there is no published proof, see [21]) that there is no(C3 ∪ C3 ∪ C5)-design of order 11. Results on the famous Oberwolfach problem providefurther G-designs of orders n for many instances where G is a 2-regular graph with n vertices,see Subsection 5.F. We also have the following three results. Theorem 5.5 was proved byBlinco and El-Zanati [33], and Theorem 5.6 was proved by Aguado et al. [19], both beingobtained under the guise of graph labelings. The first part of Theorem 5.7 is proven by Dinitzand Rodney [56] (using graph labelings) while the second part is established by Horak andRosa [77].



TABLE VI. The Spectrum for Even Graphs With up to 10 Edges

m Spectrum for each even graph with m edges Exceptions (E1 and E3 as in Fig. 3)

3 n ≡ 1 or 3 (mod 6) ∅4 n ≡ 1 (mod 8) ∅5 n ≡ 1 or 5 (mod 10) ∅6 n ≡ 1 or 9 (mod 12) There is no (C3 ∪ C3)-design of order 97 n ≡ 1 or 7 (mod 14) ∅8 n ≡ 1 (mod 16) ∅9 n ≡ 1 or 9 (mod 18) There is no G-design of order 9 for

G ∈ {C4 ∪ C5, E1, E3}10 n ≡ 1 or 5 (mod 20) and n �= 5 There is a K5-design of order 5

Theorem 5.5 ([33]). Let G be a 2-regular bipartite graph. There exists a G-design oforder n for all n ≡ 1 (mod 2|E(G)|).Theorem 5.6 ([19]). Let G be a 2-regular graph with at most three components. Thereexists a G-design of order 2|E(G)| + 1.

Theorem 5.7. Let G be the union of t vertex disjoint 3-cycles.

� There exists a G-design of order 6t + 1 [56].� If n ≡ 1 or 3 (mod 6) and n > 9t there exists a G-design of order n [77].

For m ≤ 10, the spectrum problem for each closed m-trail is completely solved in [16]and the results are included in Table VI. For m ≤ 6, the spectrum problem for closed m-trails was settled prior to [16]. When 3 ≤ m ≤ 5, the only closed m-trails are cycles (seeSection 4), and when m = 6 we have only the 6-cycle and the graph G15 (see Fig. 6). Thespectrum for G15 was settled by Bermond et al. in [28]. The spectrum problem for severalm-trails with 7 ≤ m ≤ 10 was also settled in earlier articles. For example, the spectra forG17 (a closed 7-trail, see Fig. 6) and the closed 9-trails E1 and E2, shown in Fig. 3, weresettled by Bermond et al. [28], Horak and Rosa [77], and Mullin et al. [100], respectively.There are also results on the spectrum problem for a family of closed trails known as Dutchwindmills. A Dutch l-windmill, denoted Dl, is the graph containing a vertex, x say, such thatDl is the union of l edge-disjoint 3-cycles, all of which contain the vertex x. The spectrumproblems for the Dutch 2-windmill and the Dutch 3-windmill are completely settled as theyare closed m-trails with m ≤ 10 (see Table VI). Horak and Rosa [77] proved the followingtheorem on the spectrum problem for Dutch l-windmills for arbitrary l.

Theorem 5.8 ([77]). Let Dl be the union of l edge-disjoint 3-cycles each sharing a commonvertex. If n(n − 1) ≡ 0 (mod 6l) and n > 6l then there exists a Dl-design of order n.

There are four even graphs with 10 or fewer vertices which are neither 2-regular norclosed trails, namely the graphs E3, E4, E5, and E6 shown in Figure 3. The spectra forthese four graphs are included in Table VI. The spectrum problem for E3 was partiallysolved by Horak and Rosa [77], who showed that there exists an E3-design of order n forall n ≡ 1 or 9 (mod 18) with n ≥ 55. It is easy to verify that there is no E3-design of order9 and E3-designs of orders 19, 27, 37, and 45 are given in the Appendix in Examples A.5„A.6, A.7, and A.8, respectively. These examples complete the spectrum problem for E3. It



FIGURE 3. Some even graphs.

is straightforward to check that the constructions used to settle the spectrum problem forclosed 10-trails, see Lemma 6 of [16], can also be applied to settle the spectrum problemfor E4, E5 and E6 using the decompositions given in Examples A.9–A.26 in the Appendix.

E. Theta Graphs

The theta graph �(a, b, c) is the graph consisting of three internally disjoint paths withcommon endpoints and lengths a, b, and c with a ≤ b ≤ c. The smallest theta graph is�(1, 2, 2) which has five edges. The spectrum problem for each theta graph with at mostnine edges is completely solved by Blinco in [32], and the results are summarized inTable VII.

Several instances were settled prior to [32]. Let Kn − e denote the graph on n verticesobtained from Kn by deleting an edge. We have �(1, 2, 2) ∼= K4 − e, �(1, 2, 3) ∼= G13and �(2, 2, 2) ∼= G14 (see Fig. 6). The spectra for �(1, 2, 2), �(1, 3, 3), and �(2, 2, 2)were established by Bermond and Schonheim [29], Blinco [31], and Bermond et al. [28],respectively. The spectrum problem for �(1, 2, 3) was solved in [28], with the exception ofa design of order 24. Such a design is given by Blinco [32].

The following two theorems give partial results on the spectrum problem for larger thetagraphs with specific parameters. Theorem 5.9 is due to Blinco [31] and Theorem 5.10 is a



TABLE VII. The Spectrum for Theta Graphs With up to 9 Edges

m Spectrum for theta graphs with m edges Exceptions

5 n ≡ 0 or 1 (mod 5) There is no �(1, 2, 2)-design of order 56 n ≡ 0, 1, 4 or 9 (mod 24) There is no �(2, 2, 2)-design of order 9

and n �= 4 There is no �(2, 2, 2)-design of order 127 n ≡ 0 or 1 (mod 7) There is no �(1, 3, 3)-design of order 7

There is no �(2, 2, 3)-design of order 78 n ≡ 0 or 1 (mod 16) ∅9 n ≡ 0 or 1 (mod 9) There is no �(1, 4, 4)-design of order 9

There is no �(2, 2, 5)-design of order 9There is no �(3, 3, 3)-design of order 9

consequence of results of Delorme et al. [54], Koh and Yap [88], and Punnim and Pabhapote[102] (proven under the guise of graph labelings).

Theorem 5.9 ([31]). There exists a �(1, k, k)-design of order n in each of the followingcases.

� k is odd and n ≡ 0 (mod 2k + 1) except when (k, n) = (3, 7).� k ∈ {5, 9} and n ≡ 1 (mod 2k + 1).� k ≡ 3 (mod 4) and n ≡ 1 (mod 2k + 1).� k ≡ 1 (mod 4), k ≥ 13 and n ≡ 1 (mod 4k + 2).

Theorem 5.10. Let a ≥ 1, let b, c ≥ 2, and let a ≤ b ≤ c. There exists a �(a, b, c)-designof order 2(a + b + c) + 1.

F. Unions of Graphs

In this section, we consider the existence of G-designs for cases where G is the unionof pairwise vertex-disjoint graphs, or graphs intersecting in only one vertex. Many suchresults can be obtained indirectly via solutions to other related problems, and yield onlya few scattered existence results on the spectrum problem. A description and summary ofall such results would be very lengthy, full of partial results, and not really in line withthe focus of this article. Instead, we briefly discuss a few examples to indicate the types ofresults that can be obtained in this manner.

A resolution class of a G-design D of order n is subset R of D such that in R eachvertex of Kn occurs in exactly one copy of G, and D is said to be resolvable if D can bepartitioned into resolution classes. Clearly, if t|V (G)| divides n, then the existence of aresolvable G-design of order n implies the existence of an H-design of order n where H isthe union of t vertex-disjoint copies of G. Numerous results on the existence of G-designsare thus obtained from results on the existence of resolvable G-designs. Such results havebeen obtained for complete graphs [73,103] (also see [4]), cycles [23], trees [96,120], paths[27], stars [120], cubes [18], complete bipartite graphs [81], and for K4 − e (the completegraph on four vertices with an edge removed) [51]. Also see the survey [39].

An almost resolution class of a G-design D of order n is subset R of D such that in Rthere is one vertex of Kn which occurs in none of the copies of G and each other vertexof Kn occurs in exactly one copy of G, and D is said to be almost resolvable if D can be



TABLE VIII. The Spectrum for Kk ∨ Kk (Two Copies of Kk Intersecting in a Vertex) fork ≤ 5

k Spectrum for Kk ∨ Kk Possible exceptions

2 n ≡ 0 or 1 (mod 4) ∅3 n ≡ 1 or 9 (mod 12) ∅4 n ≡ 1 or 16 (mod 24) ∅5 n ≡ 1 or 25 (mod 40) ∅

partitioned into almost resolution classes. Note that if G is a d-regular graph with d > 1 thenthere are no non-trivial almost resolvable G-designs. To see this, note that if G is a d-regulargraph with k vertices, then an almost resolvable G-design of order n has n(n−1)

dkcopies of G,

n−1k

copies of G in each almost resolution class, and hence nd

almost resolution classes. Butwe know that n − 1 ≡ 0 (mod d) is necessary for the existence of a G-design and so we seethat d = 1. Clearly, if t|V (G)| divides n − 1 and there exists an almost resolvable G-designof order n, then there exists an H-design of order n where H is the union of t vertex-disjointcopies of G. The existence problem for almost resolvable G-designs in the case G is a pathwas completely settled by Yu [119].

A k-factor of a graph K is a k-regular spanning subgraph of K and a k-factorization ofK is a set of k-factors whose edge sets partition the edge set of K. Thus, a k-factorizationof Kn in which all the k-factors are isomorphic is a G-design of Kn. In the case k = 2 sucha k-factorization is a solution to an instance of the famous Oberwolfach problem. That is,results on the Oberwolfach problem yield G-designs of Kn in the case G is a 2-regular graphon n vertices. A table of known results on the Oberwolfach problem is given in [42]. Forexample, it is known that for n ≤ 17 and for any 2-regular graph G with n vertices, there is aG-design of order n except that there is no such G-design for G ∈ {C4 ∪ C5, C3 ∪ C3 ∪ C5}[14]. There are also some results on 3-factorizations of Kn in which the 3-factors are allisomorphic [9,17]. In particular, there are 21 non-isomorphic 3-regular graphs on 10 verticesand there exists a G-design of order 10 for precisely 15 of these [17]. There are 4,207 non-isomorphic 3-regular graphs on 16 vertices and there exists a G-design of order 16 forprecisely 4,204 of these [9]. Also in [9], G-designs of order n are given for some infinitefamilies of 3-regular graphs on n vertices.

The block intersection graph for a Kk-designD of order n is the graph XD with vertex setD and edge set {B1B2 : B1, B2 ∈ D, B1 ∩ B2 �= ∅}. Various results on unions of completegraphs can be obtained from results on block intersection graphs of Kk-designs. Horak andRosa [77] show that the block intersection graph of any Kk-design of order more than kcontains a Hamilton cycle. When the number of copies of Kk is even, alternate edges ofsuch a Hamilton cycle form a perfect matching in the block intersection graph and thus wehave the following theorem.

Theorem 5.11. Let Kk ∨ Kk denote the graph consisting of two copies of Kk sharing avertex. There exists a Kk ∨ Kk-design of order n if and only if there exists a Kk-design oforder n and n(n−1)

k(k−1) is even.

Combining Theorem 5.11 with the results given in Table I, we have a complete solutionto the spectrum problem for Kk ∨ Kk when k ≤ 5, see Table VIII.

The graph consisting of two vertex-disjoint copies of Kk is denote by Kk ∪ Kk and thegraph consisting of m vertex disjoint copies of Kk by ∪mKk. Suppose there exists a Kk-



TABLE IX. The Spectrum for Kk ∪ Kk (Two Vertex-Disjoint Copies of Kk) for k ≤ 5

k Spectrum for Kk ∪ Kk Exceptions

2 n ≡ 0 or 1 (mod 4) ∅3 n ≡ 1 or 9 (mod 12) There is no (K3 ∪ K3)-design of order 94 n ≡ 1 or 16 (mod 24) ∅5 n ≡ 1 or 25 (mod 40) There is no (K5 ∪ K5)-design of order 25

design of order n and let X be its block intersection graph. A result of Hajnal and Szemeredi[71] says that for any graph Y, if m divides |V (Y )| and m <

|V (Y )|�(Y ) , where �(Y ) denotes

the maximum degree of the vertices in Y, then the vertices of Y can be partitioned intoindependent sets of size m (actually the result of [71] is stronger than this). It follows thatthe copies of Kk in a Kk-design of order n can be partitioned into copies of ∪mKk wheneverm <

v(v−1)/k(k−1)k(v−k)/(k−1) = v(v−1)

k2(v−k). The preceding argument is due to Horak and Rosa [77] who

applied it in the case k = 3. It gives us the following theorem.

Theorem 5.12. Let ∪mKk denote the graph consisting of m vertex disjoint copies of Kk.If there exists a Kk-design of order n and m <

n(n−1)k2(n−k)

, then there exists a ∪mKk design oforder n.

We can use Theorem 5.12 to settle the spectrum problem for Kk ∪ Kk when k = 4 andk = 5. The cases k = 2 and k = 3 are also settled, see Subsection 4.B or 6.A for k = 2 andSubsection 5.D for k = 3. See Table IX.

The obvious necessary conditions for the existence of a (K4 ∪ K4)-design of order nare n ≡ 1, 16 (mod 24). For n ≥ 40, we have n(n−1)

k2(n−k)> 2 and so the result follows by

Theorem 5.12 and the existence of a K4-design of order n for all n ≡ 1, 4 (mod 12). Thisleaves only the cases n = 16 and n = 25. For n = 16, it is well known that the copies of K4in the unique (up to isomorphism) K4-design of order 16 can be partitioned into 5 copiesof ∪4K4 (the design is equivalent to the affine plane of order 4) and thus into 10 copies ofK4 ∪ K4. For n = 25, see Example A.27 in the Appendix.

The obvious necessary conditions for the existence of a (K5 ∪ K5)-design of order nare n ≡ 1, 25 (mod 40). For n ≥ 65, we have n(n−1)

k2(n−k)> 2 and so the result follows by

Theorem 5.12 and the existence of a K5-design of order n for all n ≡ 1, 5 (mod 20). Thisleaves only the cases n = 25 and n = 41. For n = 25, it is well known that any K5-design oforder 25 is equivalent to an affine plane of order 5, and it follows that there is no K5 ∪ K5-design of order 25. For n = 41, see Example A.28 in the Appendix.

G. Miscellaneous

There are a few other graphs and families of graphs for which the spectrum problem hasbeen considered but which are not covered in other sections.

The Petersen graph and the Heawood graph are shown in Figure 4. The spectrum problemfor these graphs is completely solved and the results are summarized in Theorems 5.13 and5.14. Let P denote the Petersen graph. The non-existence of a P-design of order 10 isestablished by Hanson [74] (and elsewhere) and the spectrum problem for P-designs issolved in [11]. The spectrum for the Heawood graph was established in [13].



FIGURE 4. The Petersen graph and the Heawood graph.

Theorem 5.13 ([11]). Let P denote the Petersen graph. There exists a P-design of ordern if and only if n ≡ 1 or 10 (mod 15) and n �= 10.

Theorem 5.14 ([13]). Let H denote the Heawood graph. There exists an H-design of ordern if and only if n ≡ 1 or 7 (mod 21) and n �= 7.

For i ≥ 3 and m ≥ i + 1, let Di(m) denote the graph on m vertices consisting of ani-cycle and a path of m − i edges which intersect in exactly one end vertex of the path.Huang and Schonheim [84] called such graphs dragons and introduced the above notation.Theorem 5.15 is established in [84].

Theorem 5.15 ([84]). For each i ∈ {3, 4}, a Di(m)-design of order n exists if n ≡ 0 or1 (mod 2m). Moreover, this condition is also necessary when m is a power of 2.

We also have the following result on the spectrum problem for the graph obtained froma complete graph on m + 2 vertices by removing the edges of a complete subgraph on mvertices [10]. This graph is denoted by Km+2 − Km.

Theorem 5.16 ([10]). There exists a (Km+2 − Km)-design of order n for all n ≡0 or 1 (mod 2m + 1) when m is even, and for all n ≡ 1 or 2m + 1 (mod 4m + 2) when mis odd, except that there is no (Km+2 − Km)-design of order 2m + 1 when m is even.

6. SMALL GRAPHS

In this section, we consider the spectrum problem for arbitrary graphs of small order.Subsection 6.A deals with all graphs on four or fewer vertices and Subsection 6.B considersall graphs on five vertices. In Subsection 6.C, we discuss the known results for graphs withsix vertices.

A. Graphs With Four or Fewer Vertices

There are ten non-isomorphic graphs with four or fewer vertices, excluding those withisolated vertices, and these are shown in Figure 5.

The spectrum problem is completely solved for each of these graphs, with the problembeing finished off by Bermond and Schonheim in 1977 [29]. As noted in [29], the spectrumproblem for eight of the ten graphs had (effectively) already been settled. The complete



FIGURE 5. Graphs with up to 4 vertices.

TABLE X. The Spectrum for Graphs with up to 4 Vertices

Graph Spectrum

K2 all nP3 n ≡ 0 or 1 (mod 4)K3 n ≡ 1 or 3 (mod 6)K2 ∪ K2 n ≡ 0 or 1 (mod 4)P4 n ≡ 0 or 1 (mod 3) and n �= 3K1,3 n ≡ 0 or 1 (mod 3) and n �∈ {3, 4}C4 n ≡ 1 (mod 8)D3(4) n ≡ 0 or 1 (mod 8)K4 − e n ≡ 0 or 1 (mod 5) and n �= 5K4 n ≡ 1 or 4 (mod 12)

graphs K2, K3, and K4 are discussed in Section 2. The spectrum problems for the path P3and the graph K2 ∪ K2 are easy exercises, and their solutions are covered in Theorems 4.3and 4.2. The spectrum problems for C4 and K1,3 were settled, respectively, by Kotzig in1965 [90] and Cain in 1974 [46], while the spectrum problem for P4 is settled in Bermond’s1975 thesis [26]. Tarsi [111] also gives a solution for P4, see Theorem 4.3. The spectrumproblem for the remaining two graphs, namely K4 − e and D3(4), was settled in Bermondand Schonheim’s article [29]. Note that K4 − e is a �-graph, see Subsection 5.E, and D3(4)is a dragon, see Subsection 5.G. Table X gives the complete spectrum for each of the tengraphs having at most 4 vertices.

B. Graphs With Five Vertices

There are 23 non-isomorphic graphs with five vertices, excluding those with isolatedvertices, and these are shown in Figure 6.

In 1980, Bermond et al. [28] addressed the spectrum problem for each of these graphs,although the problem was (and remains) not completely solved. Table XI summarizes thecurrent state of knowledge, and includes a few updates to Theorem 24.45 of [39]. For eachgraph, the known spectrum and possible exceptions are listed. The notation is chosen tomatch that used in [28] and [39]. As we shall see, a number of anomalies have occurred



FIGURE 6. Graphs with 5 vertices.

in the literature on these spectrum problems. Thus, we give a detailed justification andexplanation of the contents of Table XI.

For i ∈ {3, 4, 5, 10, 23}, the spectrum problem for Gi was completely solved prior to1980, and Bermond et al. give the necessary references in their article [28]. We have G3 ∼=P5, G4 is a caterpillar, G5 ∼= K1,4, G10 ∼= C5, and G23 ∼= K5, and these graphs are coveredin Sections 2, 3 (especially see Theorem 3.5), and 4.

For i ∈ {1, 2, 11, 12, 13, 14, 15, 16, 17, 19} the spectrum problem for Gi is completelysettled by Bermond et al. [28], except for the unresolved cases of a G13-design of order 24and a G16-design of order n for n ∈ {119, 120, 147, 203, 204}. These cases have now allbeen settled. Blinco [32] gives a G13-design of order 24 (G13 ∼= �(1, 2, 3), see Table VII)and Chang [47] gives a G16-design for each of the above-mentioned orders. The result forG15 in Bermond et al. [28] is obtained by reference to the 1976 article by Huang [80].

We now consider the spectrum problem for Gi for i ∈ {6, 7, 8, 9}. For these valuesof i, an obvious necessary condition for the existence of a Gi-design of order n is n ≡0, 1 (mod 5). Huang [80] showed that for i ∈ {6, 7, 8, 9}, there exists a Gi-design of ordern for all n ≡ 1, 5 (mod 10), except that there is no Gi-design of order 5 for i ∈ {7, 8, 9}.For i ∈ {6, 7, 8, 9}, Bermond et al. [28] show that there exists a Gi-design of order n forall n ≡ 6 (mod 10), except that there is no G9-design of order 6. This leaves only the casen ≡ 0 (mod 10). In [28], it is claimed that this case (and also the case n ≡ 1 (mod 10)) issettled by Rosa and Huang [107]; however, it seems likely that this is a typographical error.A likely intended reference is Huang and Schonheim [84], as it is stated in [28] that G6,G7, G8, and G9 are dragons and these graphs are the topic of [84] (see also Theorem 5.15).However, the definition of dragons given in [84] excludes the graphs G6 and G7. Thus, the



TABLE XI. The Spectrum for Graphs with 5 Vertices

Graph Spectrum Possible exceptions

G1 n ≡ 0 or 1 (mod 3) and n �∈ {3, 4} ∅G2 n ≡ 0 or 1 (mod 8) ∅G3 n ≡ 0 or 1 (mod 8) ∅G4 n ≡ 0 or 1 (mod 8) ∅G5 n ≡ 0 or 1 (mod 8) ∅G6 n ≡ 0 or 1 (mod 5) ∅G7 n ≡ 0 or 1 (mod 5) and n �= 5 ∅G8 n ≡ 0 or 1 (mod 5) and n �= 5 ∅G9 n ≡ 0 or 1 (mod 5) and n �∈ {5, 6} ∅G10 n ≡ 1 or 5 (mod 10) ∅G11 n ≡ 0, 1, 4 or 9 (mod 12) and n �= 4 ∅G12 n ≡ 0, 1, 4 or 9 (mod 12) and n �= 4 ∅G13 n ≡ 0, 1, 4 or 9 (mod 12) and n �= 4 ∅G14 n ≡ 0, 1, 4 or 9 (mod 12) and n �∈ {4, 9, 12} ∅G15 n ≡ 1 or 9 (mod 12) ∅G16 n ≡ 0 or 1 (mod 7) and n �∈ {7, 8} ∅G17 n ≡ 1 or 7 (mod 14) ∅G18 n ≡ 0 or 1 (mod 7) and n �∈ {8, 14} ∅G19 n ≡ 0 or 1 (mod 7) and n �= 8 ∅G20 n ≡ 0 or 1 (mod 16) n ∈ {32, 48}G21 n ≡ 0 or 1 (mod 16) and n �= 16 n = 48G22 n ≡ 0 or 1 (mod 9) and n �∈ {9, 10, 18} n ∈ {27, 36, 54, 64, 72, 81,

90, 135, 144, 162, 216, 234}G23 n ≡ 1 or 5 (mod 20) ∅

results in Huang and Schonheim [84] settle the case n ≡ 0 (mod 10) only for G8 and G9,and the case n ≡ 0 (mod 10) remains open for G6 and G7. Although Bermond et al. [28]describe a method which may be used to settle this case, the details are not all included intheir article so we construct the necessary designs below for the sake of completeness.

Bermond et al. [28] give a G6-decomposition and a G7-decomposition of K5,5,5. Givenin Examples A.29 and A.30 are G6-designs and G7-designs of orders 10 and 20. LetKm1,m2,...,mt denote the complete multipartite graph with t parts of sizes m1, m2, . . . , mt .It is well known that for each positive integer x, there is a K3-decomposition of a completemultipartite graph of the form K2,2,...,2 or K2,2,...,2,4 and having 2x vertices (these areequivalent to certain group divisible designs with block size 3, see Ge [64]). It follows thatthere is a K5,5,5-decomposition of a complete multipartite graph of the form K10,10,...,10 orK10,10,...,10,20 and having 10x vertices. By placing a copy of a G6-design or G7-design oforder 10 or 20 on each part, and a G6-decomposition or a G7-decomposition of K5,5,5 oneach copy of K5,5,5, we obtain a G6-decomposition and a G7-decomposition of order 10x

for each positive integer x.Bermond et al. [28] completely settle the spectrum problem for G18, except that the

existence of a G18-design of order n is left unresolved for n ∈ {36, 42, 56, 92, 98, 120}.Heinrich’s survey [75] omits the case n = 42 from the list of unresolved cases. In 2004,Li and Chang [93] constructed G18-designs of order n for n ∈ {36, 56, 92, 98, 120}, andbased on the list in [75] claimed to have finished the problem. A G18-design of order 42 can



be constructed by taking the G18-decomposition of the complete multipartite graph with 6parts of size 7, given in Example A.31, and placing a G18-design of order 7 on each part.

A necessary condition for the existence of a G20-design of order n is n ≡0, 1 (mod 16). In [28], Bermond et al. construct a G20-design of order n for eachn ∈ {17, 33, 49, 97, 113, 177}. The case n ≡ 1 (mod 16) was almost completely settledin 1990 by Rodger [105], who constructed G20-designs of order n for all n ≡ 1 (mod 16)except n = 65. In 2001, Colbourn and Wan [52] constructed a G20-design of order 65, thuscompleting the solution of the n ≡ 1 (mod 16) case. Chang [47] settled the same case in2002. The first published G20-design of order n ≡ 0 (mod 16) is one of order 16 whichwas constructed by Colbourn and Wan in 2001 [52]. In [49], Colbourn et al. construct G20-designs of order n for all the other values of n ≡ 0 (mod 16) except n = 32 and n = 48.Thus, n ∈ {32, 48} are the only remaining unresolved cases for G20.

It is worth noting that [105] contains an unfortunate error in the statement of a lemma.The lemma states that the existence of G20-designs of order n for all n ≡ 0 (mod 16) isimplied by the existence of G20-designs of order 8s + 1 for various values of s, but the actualproven result requires G20-designs of order 8s. This (unproven) result is incorporated in[75] and then used in [47] to (incorrectly) establish the existence of G20-designs of order nfor all n ≡ 0 (mod 16).

A necessary condition for the existence of a G21-design of order n is n ≡ 0, 1 (mod 16).In [28], Bermond et al. construct G21-designs of order n for all n ≡ 1 (mod 16) and forn = 64, and they also show that a G21-design of order 16 does not exist. Colbourn et al. [49]construct G21-designs of order n for all n ≡ 0 (mod 16) except n = 16 (which is coveredin [28]) and n = 48. Thus, n = 48 is the only unresolved case for G21.

A necessary condition for the existence of a G22-design of order n is n ≡ 0, 1 (mod 9).Bermond et al. [28] construct a G22-design of order 19 and show that G22-designs oforder 9, 10, and 18 do not exist. Li and Chang [94] construct G22-designs of ordern for n ∈ {37, 55, 73, 109, 397, 415, 469, 487, 505, 541, 613, 685} ∪ {28, 46, 82}. Ge andLing [65], citing [28] for the solution of the cases n ∈ {9, 10, 18, 19}, have now almostcompletely settled the problem. Their result is that there exists a G22-design of ordern for all n ≡ 0, 1 (mod 9), except when n ∈ {9, 10, 18} with the possible exceptions ofn ∈ {27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, 234}.

The existence of a G22-design of order n for all n ≡ 1 (mod 18) except for twelveunresolved cases is stated in [75]. Although this result is now known to be true, a proofis not given in any of the references cited in [75]. In [94], these twelve (previously)unresolved cases are settled, but it seems the first published complete solution of the casen ≡ 1 (mod 18) is in [65].

C. Graphs With Six Vertices

The known results on the spectrum problem for graphs with six vertices concern those withat most eight edges, and a few other special graphs such as K6, the unique closed 9-trailon six vertices (E2 in Fig. 3) and the unique closed 10-trail on six vertices. The spectrumproblem for K6 is covered in Section 2 and the spectra for these two closed trails are dealtwith in Subsection 5.D.

There are 70 graphs on six vertices with at most eight edges, excluding those with isolatedvertices, and these are shown in Figure 7. Known results on the spectrum problem for eachof these graphs are given in Table XII. We use the notation H

ji for these graphs where the



FIGURE 7. Graphs with 6 vertices and up to eight edges.



TABLE XII. The Spectrum for Graphs with 6 Vertices and up to 8 Edges

(i, j) Spectrum for Hji Possible exceptions

(1, 3) n ≡ 0 or 1 (mod 3) ∅and n �∈ {3, 4}

(1, 4), (2, 4), (3, 4) n ≡ 0 or 1 (mod 8) ∅(1, 5), (2, 5), (4, 5), (8, 5), (9, 5) n ≡ 0 or 1 (mod 5) ∅

and n �= 5(3, 5), (5, 5), (6, 5), (7, 5) n ≡ 0 or 1 (mod 5) ∅

and n �= 5, 6(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6,6), n ≡ 0, 1, 4 or 9 (mod 24) ∅(7, 6), (9, 6), (10, 6), (11, 6), (12, 6), and n �= 4(13, 6), (14, 6)(8, 6) n ≡ 1 or 9 (mod 12) ∅

and n �= 9(15, 6) n ≡ 1 or 9 (mod 12) ∅(1, 7), (2, 7), (3, 7), (4, 7), (5, 7)(9, 7), n ≡ 0 or 1 (mod 7)(10, 7), (13, 7), (17, 7) ∅(6, 7), (7, 7), (12, 7), (15, 7), (18, 7) n ≡ 0 or 1 (mod 7) ∅

and n �= 7(8, 7) n ≡ 1 or 7 (mod 14) ∅(11, 7), (14, 7) n ≡ 0 or 1 (mod 7) ∅

and n �= 8(16, 7) n ≡ 0 or 1 (mod 7) n = 14t + 8

and n �= 7, 8 for all t ≥ 1(19, 7), (20, 7) n ≡ 0 or 1 (mod 7) ∅

and n �= 7, 8(1, 8), (2, 8), (3, 8), (4, 8), (5, 8), (6, 8), n ≡ 0 or 1 (mod 16) ∅(7, 8), (8, 8), (9, 8), (10, 8), (11, 8),(16, 8), (17, 8), (18, 8), (19, 8), (20, 8),(21, 8), (22, 8)(12, 8), (13, 8) n ≡ 0 or 1 (mod 16) n = 32(14, 8), (15, 8) n ≡ 1 (mod 16) ∅Designs covered by Wilson’s Theorem [116] are ignored in the listed possible exceptions for H7

16.

subscript i indexes the distinct graphs with j edges. Some of these graphs belong to specialfamilies and have thus been covered in other sections of this article. For example, some aretrees, even graphs, matchings, or theta graphs.

Yin and Gong [118] settle the spectrum problem for the 28 graphs on six vertices withat most six edges. They cite Huang and Rosa [83] for the results on H5

1 , H52 , H5

3 , H54 ,

H55 , and H5

6 (which are trees), and remark that spectrum for H615

∼= C6 is well known (seeSubsection 4.A).

The graphs H71 , H7

2 , . . . , H720 shown in Figure 7 are the 20 non-isomorphic graphs with

six vertices and seven edges, excluding those with isolated vertices. Tian et al. [112] provethe results for H7

1 , H72 , . . . , H7

16 given in Table XII. The graphs H717 and H7

18 are theta graphsand the spectrum problem for each of these had already been settled, see Subsection 5.E.It is claimed in [112] that the spectrum problem for the remaining two graphs, H7

19 andH7

20, is settled in Cui’s thesis [53]. However, as [53] is not easily accessible, we include aconstruction here. A computer search can quickly verify that there is no H7

19-design and no



H720-design of order 7 or 8. Examples A.32–A.35 and A.37–A.40 contain an H7

19-designand an H7

20-design of order n for each n ∈ {14, 15, 21, 22}. An H719-decomposition of K7,7

and an H720-decomposition of K7,7 are given in Examples A.36 and A.41, respectively. For

G ∈ {H719, H

720}, it follows easily from the existence of a G-decomposition of K7,7 that

there is a G-decomposition of any complete multipartite graph of the form K14,14,...,14 orK14,14,...,14,21. Placing a copy of a G-design of order 14 or 21 on the vertices of each partyields a G-design of order n for all n ≡ 0 (mod 7) with n ≥ 14. Adjoining a new vertex ∞and placing a copy of a G-design of order 15 or 22 on ∞ together with the vertices of eachpart yields a G-design of order n for all n ≡ 1 (mod 7) with n ≥ 15.

The graphs H81 , H8

2 , . . . , H822 shown in Figure 7 are the 22 non-isomorphic graphs with

six vertices and eight edges, excluding those with isolated vertices. Kang et al. [86] provethe results for these graphs given in Table XII. The spectrum problem for the closed trailsH8

14 and H815 had already been settled in [16].

APPENDIX

Definition. For p, q ≥ 2 we define three permutations, ρp : Zp �→ Zp, ρp,q : Zp × Zq �→Zp × Zq and ρ∞

p,q : Zp × Zq ∪ {∞} �→ Zp × Zq ∪ {∞}, as follows.

� ρp(x) = x + 1 (mod p).� ρp,q((x, y)) = (x + 1 (mod p), y).� ρ∞

p,q((x, y)) = (x + 1 (mod p), y) and ρ∞p,q(∞) = ∞.

A.1. Dodecahedron and Icosahedron Designs

The icosahedron I and dodecahedron D with vertices labeled as in Figure 8 will berepresented by (v1, v2, . . . , v12)I and (v1, v2, . . . , v20)D, respectively.

FIGURE 8. The Icosahedron and Dodecahedron.



Example A.1. Let D =

{(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)I , (0, 4, 5, 9, 8, 12, 7, 2, 13, 1, 14, 15)I ,

(0, 6, 7, 12, 10, 3, 14, 9, 11, 2, 15, 13)I , (3, 4, 6, 14, 5, 1, 13, 10, 8, 11, 15, 12)I}.

Then D is an I-design of order 16.

Example A.2. Let D consist of the orbits of the following 2 copies of D under thepermutation ρ5,5. Then D is a D-design of order 25.

((0, 0), (1, 0), (4, 4), (2, 0), (3, 4), (3, 0), (4, 3), (4, 0), (0, 1), (2, 4), (1, 4), (3, 3), (2, 1),

(3, 1), (1, 1), (0, 2), (4, 1), (1, 2), (0, 3), (2, 2))D

((0, 0), (2, 0), (3, 4), (4, 0), (1, 4), (2, 2), (2, 1), (0, 1), (0, 3), (0, 2), (1, 2), (2, 3), (1, 3),

(4, 3), (4, 2), (4, 4), (3, 3), (2, 4), (1, 1), (3, 2))D

Example A.3. Let D consist of the orbits of the following 2 copies of D under thepermutation ρ∞

13,3. Then D is a D-design of order 40.

((0, 0), (1, 0), (3, 0), (6, 0), (2, 0), (7, 0), (0, 1), (4, 0), (8, 0), (2, 1), (1, 1), (9, 0), (8, 1),

(3, 1), (4, 1), (7, 1), (9, 1), (0, 2), (1, 2), (2, 2))D

((0, 0), (4, 1), (9, 0), (1, 2), (0, 1), (9, 1), (3, 2), (2, 0), (0, 2), (5, 0), (2, 2), (7, 1), (7, 2),

(5, 2), (12, 2), (9, 2), (4, 2), (8, 2), (6, 1), ∞)D

Example A.4. Let D consist of the orbits of the following 5 copies of D under thepermutation ρ19,4. Then D is a D-design of order 76.

((10, 2), (3, 3), (2, 0), (16, 1), (14, 1), (7, 3), (8, 0), (13, 0), (18, 1), (17, 0), (6, 3), (5, 1),

(2, 1), (0, 2), (4, 1), (6, 1), (0, 3), (4, 3), (1, 0), (16, 3))D

((1, 0), (13, 3), (8, 3), (13, 2), (18, 1), (11, 2), (1, 2), (9, 0), (8, 1), (15, 2), (1, 1), (9, 3),

(10, 3), (7, 2), (0, 2), (15, 1), (6, 3), (18, 3), (15, 3), (10, 2))D

((0, 0), (1, 0), (3, 0), (6, 0), (2, 0), (8, 0), (12, 0), (4, 0), (1, 1), (5, 0), (7, 1), (9, 0), (0, 1),

(11, 0), (15, 1), (2, 2), (0, 2), (18, 1), (3, 1), (3, 2))D

((0, 0), (4, 2), (8, 0), (1, 2), (1, 0), (6, 2), (7, 2), (13, 0), (0, 2), (9, 0), (0, 3), (2, 0), (2, 3),

(8, 3), (3, 1), (2, 2), (4, 3), (15, 3), (16, 2), (6, 3))D

((0, 0), (9, 3), (0, 1), (1, 1), (2, 1), (11, 2), (15, 3), (13, 3), (0, 2), (7, 2), (11, 1), (17, 1),

(17, 2), (4, 2), (18, 3), (7, 1), (5, 3), (8, 1), (18, 2), (16, 3))D



A.2. Even Graph Designs

The graphs E3, E4, E5, and E6 with vertices labeled as in Figure 3 will be representedby (v1, v2, . . . , v8)E3 , (v1, v2, . . . , v9)E4 , (v1, v2, . . . , v9)E5 , and (v1, v2, . . . , v8)E6 ,respectively.

Example A.5. LetD consist of the orbit of (0, 1, 4, 2, 9, 3, 8, 14)E3 under the permutationρ19. Then D is an E3-design of order 19.

Example A.6. Let D consist of the orbits of the following 3 copies of E3 under thepermutation ρ∞

13,2. Then D is an E3-design of order 27.

((0, 0), (1, 0), (3, 0), (4, 0), (0, 1), (2, 0), (7, 0), (1, 1))E3

((0, 0), (6, 0), (1, 1), (2, 1), (3, 1), (1, 0), (5, 1), (7, 1))E3

((0, 1), (3, 1), (7, 1), (3, 0), (8, 1), (0, 0), (11, 1), ∞)E3

Example A.7. Let D consist of the orbits of the following 2 copies of E3 under thepermutation ρ37. Then D is an E3-design of order 37.

(0, 1, 3, 4, 9, 2, 8, 20)E3 , (0, 7, 20, 8, 22, 1, 11, 27)E3

Example A.8. Let D consist of the orbits of the following 9 copies of E3 under thepermutation ρ∞

11,4. Then D is an E3-design of order 45.

((3, 0), (7, 1), (8, 1), (2, 3), (4, 3), (1, 0), (8, 0), (1, 3))E3

((3, 3), (6, 0), (0, 2), (6, 1), (9, 2), (3, 1), (4, 3), (10, 3))E3

((6, 0), (9, 0), (7, 1), (3, 1), (9, 1), (1, 0), (10, 2), (8, 3))E3

((0, 0), (7, 1), (7, 2), (8, 2), (10, 2), (0, 2), (1, 3), (4, 3))E3

((3, 0), (5, 2), (6, 2), (8, 3), (9, 3), (2, 1), (3, 2), (0, 3))E3

((0, 0), (1, 0), (0, 1), (2, 0), (6, 2), (3, 0), (8, 0), (6, 3))E3

((0, 0), (2, 1), (6, 1), (0, 2), (2, 3), (0, 1), (3, 1), (2, 2))E3

((0, 1), (2, 1), (2, 3), (4, 2), (7, 2), (0, 2), (5, 2), (0, 3))E3

((0, 1), (5, 2), (9, 2), (6, 2), (5, 3), (0, 0), (1, 2), ∞)E3

((0, 1), (8, 2), (4, 3), (3, 3), (10, 3), (1, 1), (7, 3), ∞)E3

Example A.9. Let D consist of the orbit of (0, 2, 8, 4, 9, 6, 13, 16, 17)E4 under thepermutation ρ21. Then D is an E4-design of order 21.

Example A.10. Let D consist of the orbits of the following 6 copies of E4 under thepermutation ρ5,4. Then D is an E4-design of order 25.

((0, 0), (1, 0), (0, 1), (2, 0), (3, 1), (3, 0), (0, 2), (4, 0), (2, 2))E4

((0, 0), (2, 1), (0, 2), (0, 3), (1, 3), (1, 0), (3, 3), (4, 0), (0, 4))E4



((0, 0), (3, 3), (0, 4), (2, 4), (3, 4), (0, 1), (1, 1), (3, 1), (0, 2))E4

((0, 1), (1, 2), (4, 2), (0, 3), (2, 3), (1, 1), (4, 3), (3, 1), (0, 4))E4

((0, 1), (4, 3), (0, 4), (1, 4), (3, 4), (0, 2), (1, 2), (2, 3), (2, 4))E4

((0, 2), (0, 3), (3, 4), (2, 3), (1, 4), (2, 2), (1, 3), (3, 2), (2, 4))E4


(0, 1, 3, 4, 9, 2, 8, 15, 23)E4 , (0, 10, 23, 12, 26, 1, 17, 6, 25)E4


((6, 2), (8, 0), (3, 1), (7, 2), (7, 3), (5, 3), (4, 4), (1, 1), (5, 4))E4

((5, 0), (1, 3), (8, 4), (8, 3), (5, 4), (7, 2), (6, 3), (1, 1), (1, 4))E4

((7, 0), (3, 0), (3, 2), (6, 1), (4, 2), (5, 1), (6, 2), (2, 0), (0, 3))E4

((4, 0), (6, 1), (7, 1), (1, 4), (6, 4), (3, 3), (4, 4), (5, 2), (5, 4))E4

((0, 0), (6, 0), (1, 4), (7, 0), (8, 2), (0, 2), (3, 3), (1, 3), (4, 3))E4

((0, 0), (1, 0), (1, 1), (5, 1), (7, 1), (2, 0), (8, 1), (2, 1), (4, 2))E4

((0, 0), (3, 2), (0, 3), (1, 3), (2, 3), (1, 0), (5, 3), (8, 0), (6, 4))E4

((0, 1), (4, 1), (0, 2), (4, 2), (6, 2), (0, 0), (8, 3), (1, 1), (8, 4))E4

((0, 1), (8, 2), (1, 3), (0, 3), (5, 4), (1, 1), (3, 3), (4, 1), (7, 3))E4

((0, 1), (1, 4), (2, 4), (6, 4), (8, 4), (0, 2), (3, 2), (8, 2), (5, 4))E4

((0, 2), (7, 3), (2, 4), (1, 4), (7, 4), (1, 2), (6, 3), (2, 3), (5, 4))E4

For a graph K and a subgraph H of K, the graph obtained from K by removing the edgesof H is denoted by K − H .

Example A.13. An E4-decomposition of K25 − K5. It is easy to see that an E4-decomposition of K25 − K5 can be obtained from E4-decompositions of K10 − F and(K25 − K5) − (K10 − F ) where F is a perfect matching in the K10, and the K5 andthe K10 − F are vertex disjoint. We now construct E4-decompositions of K10 − F and(K25 − K5) − (K10 − F ).

Let D =

{(0, 2, 4, 3, 6, 1, 5, 7, 8)E4 , (5, 0, 9, 2, 6, 1, 3, 7, 4)E4 ,

(8, 3, 5, 4, 6, 1, 2, 9, 7)E4 , (9, 1, 6, 3, 4, 0, 7, 2, 8)E4}.

Then D is an E4-decomposition of K10 − F .



Let D consist of the orbits of the following 5 copies of E4 under the permutation ρ5,5.Then D is an E4-decomposition of (K25 − K5) − (K10 − F ).

((0, 0), (1, 0), (0, 1), (2, 0), (3, 1), (3, 0), (0, 2), (4, 0), (2, 2))E4

((0, 0), (2, 1), (0, 2), (0, 3), (0, 4), (1, 0), (2, 3), (3, 0), (2, 4))E4

((0, 0), (2, 3), (3, 4), (3, 3), (2, 4), (0, 1), (1, 1), (3, 1), (0, 2))E4

((0, 1), (1, 2), (1, 3), (4, 2), (0, 4), (1, 1), (0, 3), (2, 1), (3, 4))E4

((0, 1), (0, 3), (3, 4), (2, 3), (4, 4), (1, 2), (0, 4), (3, 2), (1, 4))E4

Thus, we have an E4-decomposition of K25 − K5.

Example A.14. Let D consist of the orbits of the following 6 copies of E4 under thepermutation ρ5,6. Then D is an E4-decomposition of K10,10,10.

((0, 0), (0, 2), (0, 4), (1, 2), (2, 4), (1, 0), (3, 2), (4, 0), (0, 3))E4

((0, 0), (3, 2), (1, 4), (0, 3), (3, 4), (1, 0), (3, 3), (0, 1), (4, 3))E4

((0, 1), (0, 2), (2, 4), (1, 2), (0, 4), (0, 0), (4, 4), (1, 1), (0, 5))E4

((0, 1), (2, 2), (0, 5), (3, 2), (2, 5), (0, 0), (1, 5), (4, 0), (3, 5))E4

((0, 1), (4, 2), (1, 5), (0, 3), (1, 4), (2, 2), (2, 5), (2, 3), (3, 5))E4

((0, 1), (1, 3), (3, 5), (2, 3), (4, 4), (3, 3), (3, 4), (4, 3), (2, 5))E4

Example A.15. Let D consist of the orbit of (0, 2, 8, 1, 11, 18, 6, 10, 15)E5 under thepermutation ρ21. Then D is an E5-design of order 21.


((0, 0), (1, 0), (0, 1), (2, 0), (3, 1), (2, 1), (3, 0), (1, 1), (0, 2))E5

((0, 0), (0, 2), (1, 2), (3, 2), (4, 0), (0, 3), (1, 0), (3, 3), (4, 3))E5

((0, 0), (4, 3), (0, 4), (1, 4), (3, 0), (2, 4), (0, 1), (2, 1), (0, 2))E5

((0, 1), (1, 2), (0, 3), (2, 2), (0, 2), (1, 3), (1, 1), (3, 3), (0, 4))E5

((0, 1), (3, 3), (1, 4), (4, 3), (1, 2), (0, 4), (1, 1), (3, 4), (4, 4))E5

((0, 2), (2, 3), (1, 4), (0, 3), (3, 3), (3, 4), (2, 2), (2, 4), (4, 4))E5


(0, 1, 3, 4, 9, 15, 2, 10, 19)E5 , (0, 7, 19, 10, 31, 13, 1, 12, 26)E5




((3, 3), (3, 2), (7, 3), (2, 1), (0, 2), (6, 1), (0, 1), (7, 1), (5, 3))E5

((6, 4), (3, 3), (2, 4), (4, 0), (1, 2), (5, 2), (8, 1), (4, 4), (5, 4))E5

((8, 2), (0, 0), (0, 1), (4, 1), (5, 1), (5, 3), (4, 3), (0, 4), (6, 4))E5

((6, 3), (3, 3), (7, 4), (1, 2), (5, 4), (8, 2), (6, 1), (7, 2), (6, 4))E5

((5, 0), (2, 0), (7, 1), (1, 2), (3, 2), (5, 4), (2, 1), (4, 4), (6, 4))E5

((0, 0), (1, 0), (4, 1), (2, 0), (6, 0), (7, 1), (3, 0), (0, 1), (5, 2))E5

((0, 0), (8, 1), (1, 2), (0, 2), (2, 0), (0, 3), (1, 0), (4, 2), (5, 2))E5

((0, 0), (1, 3), (2, 3), (3, 3), (4, 0), (1, 4), (1, 0), (5, 3), (7, 3))E5

((0, 0), (5, 3), (3, 4), (4, 4), (6, 0), (5, 4), (0, 1), (3, 1), (0, 2))E5

((0, 1), (4, 1), (3, 3), (2, 3), (7, 1), (8, 4), (0, 2), (6, 2), (8, 3))E5

((0, 2), (1, 3), (7, 4), (0, 4), (2, 1), (5, 4), (1, 2), (4, 3), (4, 4))E5

Example A.19.We can obtain an E5-decomposition of K25 − K5 from the following two E5-

decompositions (using the same notation and construction as in Example A.13).Let D =

{(0, 2, 4, 3, 1, 6, 5, 7, 8)E5 , (1, 2, 5, 7, 0, 9, 3, 4, 6)E5 ,

(8, 2, 6, 0, 5, 3, 4, 7, 9)E5 , (9, 5, 6, 2, 7, 3, 1, 4, 8)E5}.

Then D is an E5-decomposition of K10 − F .Let D consist of the orbits of the following 5 copies of E5 under the permutation ρ5,5.

Then D is an E5-decomposition of (K25 − K5) − (K10 − F ).

((0, 0), (1, 0), (0, 1), (2, 0), (3, 1), (2, 1), (3, 0), (1, 1), (0, 2))E5

((0, 0), (0, 2), (0, 3), (1, 2), (2, 0), (1, 3), (1, 0), (4, 2), (0, 4))E5

((0, 0), (2, 3), (0, 4), (1, 4), (0, 1), (2, 4), (1, 0), (4, 3), (4, 4))E5

((0, 1), (2, 1), (0, 3), (0, 2), (0, 4), (1, 2), (1, 1), (2, 3), (1, 4))E5

((0, 1), (2, 2), (4, 4), (3, 2), (1, 4), (4, 3), (4, 1), (1, 3), (2, 4))E5


((1, 2), (0, 1), (0, 4), (0, 5), (1, 0), (4, 5), (4, 1), (0, 3), (3, 4))E5

((3, 2), (1, 1), (3, 5), (0, 5), (0, 3), (4, 5), (4, 0), (1, 2), (1, 4))E5



((0, 0), (0, 2), (1, 4), (1, 2), (2, 0), (0, 3), (1, 0), (2, 3), (4, 4))E5

((0, 0), (3, 2), (0, 4), (2, 3), (2, 1), (1, 5), (0, 1), (4, 2), (2, 4))E5

((0, 0), (4, 3), (2, 5), (4, 4), (3, 3), (0, 5), (0, 1), (2, 3), (1, 4))E5

((0, 1), (3, 3), (3, 4), (0, 2), (2, 1), (3, 5), (1, 1), (0, 3), (1, 5))E5

Example A.21. Let D consist of the orbit of (0, 1, 3, 5, 10, 2, 8, 15)E6 under thepermutation ρ21. Then D is an E6-design of order 21.


((0, 0), (1, 0), (0, 1), (2, 1), (0, 2), (2, 0), (4, 0), (0, 3))E6

((0, 0), (3, 1), (1, 2), (2, 2), (3, 2), (1, 0), (0, 3), (1, 3))E6

((0, 0), (2, 3), (0, 4), (1, 4), (2, 4), (0, 1), (1, 1), (2, 2))E6

((0, 1), (2, 1), (0, 3), (1, 3), (0, 4), (0, 2), (2, 2), (2, 3))E6

((0, 2), (3, 3), (4, 2), (1, 3), (4, 4), (2, 2), (0, 4), (2, 4))E6

((0, 4), (1, 4), (2, 0), (4, 1), (4, 2), (0, 1), (2, 3), (4, 4))E6


(0, 1, 3, 5, 7, 2, 10, 20)E6 , (0, 9, 21, 22, 24, 1, 12, 26)E6


((7, 1), (8, 1), (0, 2), (4, 2), (8, 4), (7, 0), (4, 3), (1, 4))E6

((6, 0), (4, 1), (1, 1), (2, 2), (5, 3), (8, 2), (2, 4), (5, 4))E6

((7, 1), (2, 3), (1, 4), (5, 4), (6, 4), (4, 2), (5, 2), (8, 3))E6

((2, 0), (4, 4), (5, 0), (6, 0), (2, 2), (0, 1), (5, 1), (5, 3))E6

((2, 1), (8, 3), (8, 0), (2, 2), (0, 3), (1, 1), (3, 3), (3, 4))E6

((0, 0), (1, 0), (0, 1), (2, 1), (6, 1), (2, 0), (4, 0), (1, 2))E6

((0, 0), (1, 2), (3, 2), (4, 2), (1, 3), (1, 0), (8, 2), (4, 3))E6

((0, 0), (2, 2), (4, 3), (0, 4), (1, 4), (1, 0), (3, 3), (6, 3))E6

((0, 1), (4, 4), (7, 1), (3, 2), (5, 4), (2, 0), (6, 4), (8, 4))E6

((0, 2), (8, 3), (1, 3), (0, 4), (4, 4), (0, 1), (4, 2), (8, 2))E6

((0, 3), (7, 4), (2, 0), (2, 2), (2, 4), (0, 1), (3, 3), (8, 3))E6



Example A.25. An E6-decomposition of K25 − K5. Using the same notation as inExample A.13, it is easy to see that an E6-decomposition of K25 − K5 can be obtained fromE6-decompositions of K15 − (C5 ∪ C5 ∪ C5) and (K25 − K5) − (K15 − (C5 ∪ C5 ∪ C5))where the K5 and the K15 − (C5 ∪ C5 ∪ C5) are vertex disjoint. We now construct E6-decompositions of K15 − (C5 ∪ C5 ∪ C5) and (K25 − K5) − (K15 − (C5 ∪ C5 ∪ C5)).

Let D =

{(0, 2, 5, 6, 7, 1, 3, 8)E6 , (0, 3, 9, 10, 11, 1, 4, 5)E6 , (0, 8, 12, 13, 14, 1, 6, 9)E6 ,

(1, 7, 10, 11, 13, 2, 4, 8)E6 , (2, 9, 10, 11, 13, 3, 6, 12)E6 , (3, 5, 7, 13, 14, 6, 8, 10)E6 ,

(4, 7, 9, 12, 14, 5, 8, 11)E6 , (6, 11, 4, 13, 14, 5, 10, 12)E6 , (12, 14, 1, 2, 9, 4, 10, 13)E6}.

Then D is an E6-decomposition of K15 − (C5 ∪ C5 ∪ C5).Let D consist of the orbits of the following 4 copies of E6 under the permutation ρ5,5.

Then D is an E6-decomposition of (K25 − K5) − (K15 − (C5 ∪ C5 ∪ C5)).

((1, 2), (2, 0), (4, 4), (4, 0), (0, 2), (0, 1), (1, 1), (3, 0), (2, 4))E6

((0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (3, 0), (4, 1), (0, 4), (0, 3))E6

((0, 0), (2, 1), (0, 4), (0, 3), (1, 3), (4, 1), (3, 4), (2, 3), (4, 4))E6

((0, 0), (3, 3), (1, 4), (4, 3), (3, 4), (0, 2), (0, 4), (1, 2), (2, 4))E6

Thus, we have an E6-decomposition of K25 − K5.


((3, 0), (4, 5), (1, 2), (3, 2), (2, 3), (0, 1), (2, 2), (3, 4))E6

((1, 3), (0, 5), (0, 0), (1, 1), (2, 1), (4, 1), (2, 3), (3, 4))E6

((0, 0), (4, 2), (1, 4), (2, 4), (4, 4), (1, 0), (3, 3), (3, 5))E6

((0, 1), (1, 3), (0, 4), (1, 4), (2, 5), (0, 0), (0, 3), (3, 4))E6

((0, 1), (0, 5), (0, 2), (1, 2), (2, 3), (0, 0), (3, 3), (0, 4))E6

((0, 2), (2, 5), (3, 0), (4, 0), (1, 1), (0, 1), (3, 2), (2, 4))E6

A.3. Designs for Unions of Graphs

The graph Kk ∪ Kk with vertices v1, v2, . . . , v2k and edge set

{vivj : 1 ≤ i < j ≤ k} ∪ {vivj : k + 1 ≤ i < j ≤ 2k}

is represented by (v1, v2, . . . , vk : vk+1, vk+2, . . . , v2k)Kk∪Kk.



Example A.27. Let D consist of the orbits of the following 5 copies of K4 ∪ K4 under thepermutation ρ5,5. Then D is a (K4 ∪ K4)-design of order 25.

((0, 0), (1, 0), (0, 1), (4, 4) : (2, 0), (1, 3), (2, 4), (3, 4))K4∪K4

((0, 0), (2, 0), (0, 2), (3, 3) : (1, 1), (3, 1), (1, 3), (4, 4))K4∪K4

((0, 0), (1, 1), (2, 1), (1, 2) : (3, 1), (0, 2), (4, 2), (4, 3))K4∪K4

((0, 0), (3, 1), (0, 3), (2, 3) : (0, 1), (3, 2), (0, 4), (2, 4))K4∪K4

((0, 0), (2, 2), (4, 2), (2, 4) : (0, 2), (1, 3), (2, 3), (1, 4))K4∪K4

Example A.28. LetD consist of the orbit of (0, 1, 4, 11, 29 : 4, 6, 12, 21, 26)K5∪K5 underthe permutation ρ41. Then D is a (K5 ∪ K5)-design of order 41.

A.4. Designs for Graphs of Order 5

The graphs G6, G7, and G18 with vertices labeled as in Figure 9 will be represented by(v1, v2, . . . , v5)G6 , (v1, v2, . . . , v5)G7 , and (v1, v2, . . . , v5)G18 , respectively.

Example A.29. Let D1 consist of the orbit of (0, 1, 5, 8, 3)G6 under the permutation ρ10and let D2 consist of the orbit of (3, 0, 4, 5, 1)G7 under the permutation ρ10. Then D1 is aG6-design of order 10 and D2 is a G7-design of order 10.

Example A.30. Let D1 consist of the orbits of (1, 0, 7, 11, 3)G6 and (13, 9, 15, 18, 8)G6

under the permutation ρ20 and let D2 consist of the orbits of (1, 0, 7, 8, 3)G7 and(13, 9, 15, 19, 18)G7 under the permutation ρ20. Then D1 is a G6-design of order 20 andD2 is a G7-design of order 20.

Example A.31. Let D consist of the orbits of the following 5 copies of G18 under thepermutation ρ21,2. ThenD is a G18-decomposition of K7,7,7,7,7,7. The six parts of K7,7,7,7,7,7are {(i, j), (i + 3, j), (i + 6, j), . . . , (i + 18, j)} for i ∈ {0, 1, 2} and j ∈ {0, 1}.

((0, 0), (1, 0), (3, 0), (8, 0), (0, 1))G18 , ((0, 0), (4, 0), (12, 0), (0, 1), (1, 2))G18

((0, 0), (2, 1), (3, 0), (7, 1), (11, 1))G18 , ((0, 0), (7, 1), (16, 0), (9, 1), (19, 1))G18

{((0, 1), (7, 1), (2, 0), (13, 0), (8, 1))G18

FIGURE 9. The graphs G6, G7 and G18



FIGURE 10. The graphs H719 and H7

20

A.5. Designs for Graphs of Order 6

The graphs H719 and H7

20 with vertices labeled as in Figure 10 will be represented by(v1, v2, . . . , v6)H7

19and (v1, v2, . . . , v6)H7

20, respectively.

Example A.32. Let D consist of the orbit of (2, 0, 6, 13, 1, 4)H719

under the permutation

ρ14. Then D is an H719-design of order 14.

Example A.33. Let D consist of the orbits of the following 3 copies of H719 under the

permutation ρ5,3. Then D is an H719-design of order 15.

((0, 0), (1, 0), (3, 0), (0, 1), (1, 1), (0, 2))H719

, ((0, 0), (0, 1), (1, 0), (1, 2), (2, 2), (4, 1))H719

((0, 1), (2, 1), (2, 2), (4, 0), (1, 2), (4, 2))H719



((0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (1, 1))H719

, ((0, 0), (1, 2), (2, 0), (0, 3), (2, 3), (0, 1))H719

((0, 0), (0, 4), (1, 0), (2, 4), (0, 5), (0, 1))H719

, ((0, 0), (1, 5), (0, 1), (1, 1), (0, 6), (2, 1))H719

((0, 0), (1, 6), (0, 2), (1, 2), (2, 6), (0, 3))H719

, ((0, 1), (0, 3), (2, 1), (1, 6), (1, 4), (0, 2))H719

((0, 2), (1, 3), (2, 2), (1, 4), (0, 5), (0, 3))H719

, ((0, 2), (0, 4), (0, 3), (1, 4), (2, 5), (2, 4))H719

((0, 3), (2, 4), (0, 5), (2, 3), (0, 6), (2, 6))H719

, ((0, 5), (1, 5), (0, 6), (0, 2), (1, 4), (2, 6))H719



((0, 0), (1, 0), (3, 0), (6, 0), (0, 1), (5, 0))H719

, ((0, 0), (5, 0), (1, 1), (2, 0), (2, 1), (6, 1))H719

((0, 1), (2, 1), (5, 1), (1, 0), (2, 0), (7, 1))H719

Example A.36. Let D consist of the orbit of ((0, 1), (0, 0), (2, 1), (3, 0), (4, 0), (1, 1))H719

under the permutation ρ7,2. Then D is an H719-decomposition of K7,7.



Example A.37. Let D consist of the orbit of (2, 0, 6, 8, 1, 4)H720

under the permutation

ρ14. Then D is an H720-design of order 14.



((0, 0), (1, 0), (3, 0), (2, 0), (0, 1), (2, 1))H720

, ((0, 0), (4, 1), (0, 1), (1, 1), (0, 2), (1, 2))H720

((0, 0), (1, 2), (4, 0), (0, 2), (3, 2), (0, 1))H720



((0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (1, 1))H720

, ((0, 0), (1, 1), (0, 1), (1, 0), (0, 3), (2, 2))H720

((0, 0), (2, 2), (0, 2), (1, 0), (0, 4), (1, 3))H720

, ((0, 0), (1, 3), (0, 1), (1, 0), (0, 5), (2, 1))H720

((0, 0), (1, 4), (0, 1), (1, 0), (0, 6), (1, 1))H720

, ((0, 0), (1, 5), (0, 2), (0, 1), (1, 6), (1, 2))H720

((0, 1), (2, 4), (0, 2), (0, 3), (2, 5), (1, 3))H720

, ((0, 2), (0, 3), (1, 3), (2, 3), (2, 6), (0, 4))H720

((0, 3), (0, 5), (1, 4), (0, 6), (2, 6), (2, 5))H720

, ((0, 4), (0, 6), (1, 5), (0, 2), (1, 4), (2, 5))H720



((0, 0), (1, 0), (3, 0), (2, 0), (6, 0), (0, 1))H720

((0, 0), (0, 1), (2, 0), (1, 0), (3, 1), (7, 0))H720

((0, 0), (6, 1), (0, 1), (7, 1), (8, 1), (4, 1))H720

Example A.41. Let D consist of the orbit of ((0, 1), (0, 0), (2, 1), (3, 1), (4, 0), (1, 1))H720

under the permutation ρ7,7. Then D is an H720-decomposition of K7,7.

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