One-diregular subgraphs in semicomplete multipartite digraphs

One-Diregular Subgraphs inSemicomplete MultipartiteDigraphs

Anders YeoDEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE,

ODENSE UNIVERSITY, DENMARKe-mail: [email protected]

ABSTRACT

The problem of finding necessary and sufficient conditions for a semicomplete multi-partite digraph (SMD) to be Hamiltonian, seems to be both very interesting and difficult.Bang-Jensen, Gutin and Huang (Discrete Math to appear) proved a sufficient conditionfor a SMD to be Hamiltonian. A strengthening of this condition, shown in this paper,allows us to prove the following three results. We prove that every k-strong SMD withat most k-vertices in each color class is Hamiltonian and every k-strong SMD has a cyclethrough any set of k vertices. These two statements were stated as conjectures byVolkmann (L. Volkmann, a talk at the second Krakw Conference of Graph Theory (1994))and Bang-Jensen, Gutin, and Yeo (J. Bang-Jensen, G. Gutin, and A. Yeo, On k-strong andk-cyclic digraphs, submitted), respectively. We also prove that every diregular SMD isHamiltonian, which was conjectured in a weaker form by Zhang (C.-Q. Zhang, Hamiltonpaths in multipartite oriented graphs, Ann Discrete Math. 41 (1989), 499–581). c© 1997John Wiley & Sons, Inc.

1. INTRODUCTION

This paper deals with cycles in semicomplete multipartite digraphs (SMD's). A SMD (or semi-complete k-partite digraph) is a digraph obtained from a complete k-partite graph by substitutingeach edge with an arc, or a pair of mutually opposite arcs with the same end vertices. Specialcases of SMD's are semicomplete bipartite digraphs (k = 2) and semicomplete digraphs (k = n,the number of vertices).

Bang-Jensen, Gutin and Huang proved a sufficient condition for a SMD to be Hamiltonian(see [3]). A strengthening of this condition, shown in this paper, allows us to prove the following.

Journal of Graph Theory Vol. 24, No. 2, 175 185 (1997)c© 1997 John Wiley & Sons, Inc. CCC 0364-9024/97/020175-11

176 JOURNAL OF GRAPH THEORY

Let D be a (bk/2c + 1)-strong SMD, and let X be an arbitrary set of vertices in D, with at mostk vertices in each color class. If there is a number of disjoint cycles covering X , then there is acycle covering X . This result, together with the fact that every k-strong SMD has a collection ofdisjoint cycles covering any set of vertices, with at most k vertices in each color class, implies thefollowing. Every k-strong SMD has a cycle covering any set of vertices, with at most k verticesin each color class.

This theorem provides an affirmative answer to a conjecture by Guo and Volkmann [10] thatevery k-strong SMD, with at most k vertices in each color class, is Hamiltonian and a conjectureby Bang-Jensen, Gutin and Yeo that there is a cycle through any set of k vertices in a k-strongSMD. We also prove that every diregular SMD is Hamiltonian. This was conjectured for diregularSMD's without 2-cycles by Zhang [11].

2. TERMINOLOGY AND NOTATION

We assume that the reader is familiar with the standard terminology on graphs and digraphs andrefer the reader to [5].

A digraph D = (V (D), A(D)) is determined by its set of vertices, V (D), and its set of arcs,A(D). An arc in D is an ordered pair of vertices. If xy ∈ A(D)(x, y ∈ V (D)), then we say thatthere is an arc from x to y in D, and that x dominates y and y is dominated by x. We also denotethis by x → y. If X, Y ⊆ V (D) and there is no arc from Y to X , then we say that X ⇒ Y (calledX strongly dominates Y ). Sometimes, when no ambiguity exists, we shall write X ⇒ Y , whenwe actually mean V (X) ⇒ V (Y ). For a vertex x ∈ V (D) the in-degree, d−(x) (out-degree,d+(x)) of x is the number of vertices dominating x (dominated by x) in D. By a cycle (path)we mean a simple directed cycle (path, respectively). A Hamilton cycle of D is a cycle, whichcontains all vertices of D. If Q is a subgraph in a digraph D, then D〈Q〉 is the subgraph inducedby the vertices in Q.

For two subgraphs X, Y of D, a path P is called an (X, Y )-path if P starts at a vertexx ∈ V (X), terminates at a vertex y ∈ V (Y ) and V (P ) ∩ V (X ∪ Y ) = {x, y}. If X = {x} andY = {y}, we shall say that P is an (x, y)-path. The (v, w)-subpath of a path or cycle Q will bedenoted by Q[v, w].

If x is a vertex on a cycle, C, then x+(x−) denotes the sucessor (predecessor) of x on C. IfC is a cycle in D, then the internal arcs of C are D〈C〉 − A(C) (i.e., the arcs that have bothend-vertices in V (C), but can't be written on the form xx+).

A digraph, D, is strong if it contains an (x, y)-path and an (y, x)-path for all pairs of verticesx, y in D. A digraph, D, is k-strong if for any S ⊂ V (D) with |S| < k, D − S is strong. A1-diregular subgraph, F , of D, is a collection of disjoint cycles in D. A 1-diregular subgraphF of D is called a factor if V (F ) = V (D). A minimal factor of D is a factor that contains theminimum number of cycles. For example a Hamilton cycle will always be a minimal factor.

Let D be a digraph with a 1-diregular subgraph F = C1 ∪ C2 ∪ · · · ∪ Ct (Ci is a cycle inF, i = 1, 2, . . . , t). We shall say that P = p1p2 · · · pl is a (Ci, Cj)F -path, if V (F ) ∩ V (P ) ={p1, pl} and P is a (Ci, Cj)-path. We shall say that a (Ci, Cj)-path P is a (Ci, Cj)◦

F -path, if foreach cycle Cm ∈ F − Ci − Cj , either V (Cm) ⊆ V (P ) or V (Cm) ∩ V (P ) = ∅.

A semicomplete multipartite digraph (SMD) is obtained by replacing each edge of a completek-partite graph (k ≥ 2) with an arc, or a pair of mutually opposite arcs with the same end vertices.A SMD is a multipartite tournament if it has no cycles of length 2. Whenever we consider a SMD,D, we shall use the term ‘color classes’ to denote the uniquely determined partition classes ofV (D). If x ∈ V (D), where D is a SMD, then we denote the color class which x belongs to by

ONE-DIREGULAR SUBGRAPHS 177

V c(x), so if x and y are not adjacent in a SMD, D, then V c(x) = V c(y), and if they are adjacentthen V c(x) /= V c(y).

A diregular digraph is a digraph where for each vertex v d+(v) = d−(v) = k, for someconstant k (not depending on v).

Let P be an (x, y)-path in a digraph D and let Q = v1v2 · · · vl be a path or cycle in D−V (P ).Then we say that P has a partner on Q if there is an arc (the partner of P ) vi → vi+1 on Q suchthat vi → x and y → vi+1. In this case the path, P , can be inserted in Q to give a new path (orcycle) Q[v1, vi]PQ[vi+1, vl]. We will often consider partners for paths of length 0 or 1, i.e., forvertices and arcs.

3. MAIN RESULTS

In this section we will show some properties of 1-diregular subgraphs, which cover a set ofvertices, with the minimum number of cycles. The next lemma is similar to Lemma 3.2 in [3],and it is a special case of Lemma 3.1 in [4] (see also [7]).

Lemma 3.1. Let T be a digraph. Suppose that P = p1p2 · · · pl is a path in T , and C is a cyclein T − V (P ). Suppose that, for each odd i, pi → pi+1 has a partner on C, and if l is odd, thenpl has a partner on C. Then T contains a cycle with vertex set V (P ) ∪ V (C).

Definition 3.2. Let D be a digraph, with two disjoint cycles C1 and C2. We shall write C1 '>C2 when the following is true. There is a vertex x1 ∈ V (C1), such that x1 ⇒ V (C2), and thereis no vertex y1 ∈ V (C1), such that V (C2) ⇒ y1. Furthermore there is a vertex x2 ∈ V (C2),such that V (C1) ⇒ x2, and there is no vertex y2 ∈ V (C2), such that y2 ⇒ V (C1).

The following lemma is equivalent to Lemma 4.3 in [3].

Lemma 3.3. Let D be a SMD, with a 1-diregular subgraph, F = C1 ∪ C2 ∪ · · · ∪ Ct, suchthat there is no other 1-diregular subgraph, F ′, consisting of fewer cycles than F , with V (F ) =V (F ′). For any i and j, with 1 ≤ i /= j ≤ t, we have that either Ci '> Cj or Cj '> Ci (butnot both).

Lemma 3.4. Let D be a SMD, and let C1 and C2 be two disjoint cycles in D, such that C1 '> C2and C1 /⇒ C2. Assume that there is no cycle in D, with vertex set V (C1) ∪ V (C2). Then thereexists a unique color class V of D such that for any (C2, C1)-path, P = p1p2 · · · pl, in D, either{p+

1 , p−l } ⊆ V or there exists a cycle C∗ in D, with V (C∗) = V (C1) ∪ V (C2) ∪ V (P ).

Proof. Since C1 '> C2 and C1 /⇒ C2, there is a vertex x ∈ V (C1), with x ⇒ C2 andx+ /⇒ C2. Let V = V c(x). Let y ∈ V (C2) be choosen such that y− → x+. The vertexy ∈ V , since otherwise C = C2[y, y−]C1[x+, x]y is a cycle with V (C) = V (C1) ∪ V (C2). Wewill now show the following three statements (i), (ii) and (iii). The Claims (ii) and (iii), whereP = p1p2 · · · pl is an arbitrary (C2, C1)-path, imply the lemma.

(i) C1 ⇒ y: Label the vertices in C2 such that C2 = y1y2 . . . ymy1, where y1 = y, andassume that statement (i) is not true, i.e., C1 /⇒ y1. Now define the statements αK and βK

as follows.

αK : The vertex yk ∈ V and C1 /⇒ yk, for all k = 1, 3, 5, . . . , K.

βK : The arc yky+k has a partner in C1[x+, x], for all k = 1, 3, 5, . . . , K.


We will now show that αK and βK are true for all odd K, with 1 ≤ K < m. Clearly α1holds, so if we prove the following two operations, we are done by induction.

αK and βK−2 imply βK (when K = 1, αK implies βK): If yK has a partner in C1, then itmust also have a partner in C1[x+, x], since xx+ cannot be a partner of yK , because byαK , yK ∈ V c(x). By βK−2 and Lemma 3.1 we can insert the path C2[y1, yK ] into thecycle C1[x+, x]C2[y+

K , ym]x+. So we may assume that yK has no partner in C1. SinceC1 /⇒ yK , there must be a zK ∈ V (C1) such that zK ∈ V and z−

K → yK → z+K . Now

y+K → zK , since there otherwise would be a cycle, C = C2[y+

K , yK ]C1[z+K , zK ]y+

K , inD with V (C) = V (C1) ∪ V (C2). Thus yKy+

K has the partner z−KzK , which implies

that yKy+K has a partner in C1[x+, x], since z−

K /= x(z−K /∈ V ).

αK−2 and βK−2 imply αK : yK ∈ V , since otherwise by βK−2 and Lemma 3.1 we caninsert the path P = y1y2 · · · yK−1 into the cycle C1[x+, x]C2[yK , ym]x+ and obtain acycle in D with vertex set V (C1) ∪ V (C2). If C1 ⇒ yK , then z−

K−2 → yK , wherezK−2 was defined when we proved βK−2. When we defined zK−2, we found thaty+

K−2 → zK−2. The cycle C = C1[zK−2, z−K−2]C2[yK , y+

K−2]zK−2, has V (C) =V (C1) ∪ V (C2), a contradiction. This completes the proof that αK holds.

Since ym has a partner in C1 (namely xx+), Lemma 3.1 implies that we can insert the pathC2[y1, ym] into C1 to obtain a new cycle in D with vertex set V (C1) ∪ V (C2). This is acontradiction, which implies that C1 ⇒ y.

(ii) If {p+1 , p−

l } ∩ V = ∅, then the cycle C∗ = C1[x+, p−l ]C2[y, p1]PC1[pl, x]C2[p+

1 , y−]x+,has V (C∗) = V (C1) ∪ V (C2) ∪ V (P ).

(iii) If V c(p+1 ) /= V c(p−

l ), then D contains a cycle C∗, with V (C∗) = V (P )∪V (C1)∪V (C2).Assume that C∗ does not exist. Claim (ii) implies that either p+

1 ∈ V or p−l ∈ V , but

not both, since V c(p+1 ) /= V c(p−

l ). Now we may assume that p+1 → p−

l , since otherwiseC∗ = C1[pl, p

−l ]C2[p+

1 , p1]P would have V (C∗) = V (P ) ∪ V (C1) ∪ V (C2). NowClaim (ii), used for the path P ′ = p+

1 p−l , implies that either p++

1 ∈ V or p−−l ∈ V ,

but not both, since either p+1 ∈ V or p−

l ∈ V . Continuing this process we obtain thatp+1 → p−

l , p++1 → p−−

1 , · · · which clearly is impossible since C1 has a vertex whichstrongly dominates C2. This is a contradiction, hence C∗ exists.

From now on we assume that D is a SMD, X ⊆ V (D) is arbitrary and F is a 1-diregularsubgraph of D consisting of t cycles, that covers X , such that t is minimum. We will prove someproperties of F .

Theorem 3.5. The cycles in F can be labeled in a unique way C1, C2, . . . , Ct, such that Ci '>Cj , for all i, j satisfying 1 ≤ i < j ≤ t.

Proof. Define a digraph T , with vertex set {C1, C2, . . . , Ct}, such that Ci → Cj in T , if andonly if Ci '> Cj . By Lemma 3.3, T is a tournament. This theorem states that T is transitive, soall we have to prove is that there are no cycles of length 3 in T .

Assume that T has a cycle of length 3, and w.l.o.g that C1 → C2 → C3 → C1, is a cycleof length 3 in T . If there are two cycles Ci and Cj , with (i, j) ∈ {(1, 2), (2, 3), (3, 1)}, suchthat Ci /⇒ Cj , then assume w.l.o.g that i = 1 and j = 2. By Lemma 3.4 and the minimalityof t, there is a unique color-class, V , such that any (C2, C1)◦

F -path, P = p1p2 · · · pl, in D, has{p+

1 , p−l } ∈ V . Since C2 '> C3 '> C1, there exists wk ∈ V (Ck), k = 1, 2, 3, such that


w2 ∈ V, w2 → w3 and w−3 → w1. Now P = w2C3[w3, w

−3 ]w1 is a (C2, C1)◦

F -path in D, withw+

2 /∈ V , which is a contradiction against Lemma 3.4.If C1 ⇒ C2 ⇒ C3 ⇒ C1, then, by Lemma 5.2 in [3], D〈C1 ∪ C2 ∪ C3〉 is Hamiltonian. This

is a contradiction against the minimality of t.

By Theorem 3.5 we may now assume that the cycles C1, . . . , Ct of F are ordered, such thatfor every i and j, with 1 ≤ i < j ≤ t, we have that Ci '> Cj .

Theorem 3.6. Every (Cj , Ci)◦F -path, P = p1p2 · · · pl, in D, with 1 ≤ i < j ≤ t has

{p+1 , p−

l } ⊆ X ∩ V c(p+1 ).

Proof. Let P = p1p2 · · · pl be any (Cj , Ci)◦F -path in D (1 ≤ i < j ≤ t).

Suppose that Ci /⇒ Cj , then by Lemma 3.4 there is a unique color class, Vij , such that{p+

1 , p−l } ⊆ Vij . Assume that p+

1 /∈ X . If p−l → p++

1 then the cycle C∗ = Ci[pl, p−l ]Cj [p++

1 , p1]×P , with V (C∗)∩X ⊆ V (Ci)∪V (Cj)∪V (P ) contradicts the minimality of t. If p++

1 → p−l ,

then since p−l /∈ Vij , Lemma 3.4 implies that we can construct a cycle, C∗, with V (C∗) =

V (Ci) ∪ V (Cj), which is a contradiction against the minimality of t. We have shown thatp+1 ∈ X and the claim p−

l ∈ X can be proved similarly.Suppose that Ci ⇒ Cj . If V c(p+

1 ) /= V c(p−l ) then the cycle Ci[pl, p

−l ]Cj [p+

1 , p1]P is acontradiction against the minimality of t, so we may assume that V c(p+

1 ) = V c(p−l ). Now if

p+1 /∈ X , then the cycle Ci[pl, p

−l ]Cj [p++

1 , p1]P is a contradiction against the minimality of F ,so we must have p+

1 ∈ X . Analogously we must have p−l ∈ X .

We have now shown that {p+1 , p−

l } ⊆ X ∩ V c(p+1 ), whether or not Ci ⇒ Cj .

Theorem 3.7. For every i and j, with 1 ≤ i < j ≤ t, there exists a color class V c∗ (i, j), such

that every (Cj , Ci)F -path, P = p1p2 · · · pl, in D has {p+1 , p−

l } ⊆ X ∩ V c∗ (i, j).

Proof. Let P = p1p2 · · · pl and Q = q1q2 · · · qm be arbitrary (Cj , Ci)F -paths (1 ≤ i <j ≤ t). By Theorem 3.6 all we have to prove is that V c(p+

1 ) = V c(q+1 ), since we then can set

V c∗ (i, j) = V c(p+

1 ).Assume that V c(p+

1 ) /= V c(q+1 ), which by Theorem 3.6 implies that V c(p−

l ) = V c(p+1 ) /=

V c(q+1 ) = V c(q−

m). If Ci /⇒ Cj , then using Lemma 3.4 we obtain a contradiction against theminimality of t. Therefore we may assume that Ci ⇒ Cj .

If P and Q are not disjoint, then let x ∈ V (P ) ∩ V (Q) be the first vertex on Q which liesin V (P ∩ Q). The (Cj , Ci)◦

F -path, R = Q[q1, x]P [x, pl] = r1r2 · · · rn, now has V c(r+1 ) /=

V c(r−n ), which is a contradiction against Theorem 3.6.

If P and Q are disjoint, then the cycle QCi[qm, p−l ]Cj [q+

1 , p1]PCi[pl, q−m]Cj [p+

1 , q1] is acontradiction against the minimality of t.

Theorem 3.8. Let V1, V2, . . . , Vc be the color classes of D. There exist integers 1 = n0 < n1 <· · · < nm = t and q1, . . . , qm ∈ {1, 2, . . . , c}, such that the following holds. For every (Cj , Ci)F -path,P = p1p2 · · · pl , in D with 1 ≤ i < j ≤ t, there exists an integer k ∈ {1, 2, . . . , m}, suchthat nk−1 ≤ i < j ≤ nk and {p+

1 , p−l } ⊆ Vqk

∩ X .

Proof. Define the set, N = {n0, n1, . . . , nm} with 1 = n0 < n1 < · · · < nm = t, in thefollowing way.

N = {n|2 ≤ n < t there is no (Cn+1 ∪ · · · ∪ Ct, C1 ∪ · · · ∪ Cn−1)F -path in D} ∪ {1, t}Let P = p1p2 · · · pl be a (Cj , Ci)F -path in D with 1 ≤ i < j ≤ t. From the definition of

n0, n1, . . . , nm, there must exist an integer k ∈ {1, 2, . . . , m}, such that nk−1 ≤ i < j ≤ nk.To prove the theorem, we need to show that there is a color-class Vqk

(which does not depend on


P ), such that {p+1 , p−

i } ⊆ Vqk∩ X . If nk = nk−1 + 1, then we are done immediately, by setting

Vqk= V c

∗ (nk−1, nk), according to Theorem 3.7, so assume that nk > nk−1 + 1.From the definition of N , we get that there exist integers a0, a1, . . . , au, b1, b2, . . . , bu (u ≥ 1)

and (not necessarily disjoint) (Cas , Cbs)F -paths, Ps, for s = 1, 2, . . . , u, such that nk−1 + 1 =a0 < a1 < · · · < au = nk and nk−1 ≤ br < ar−1 for r = 1, 2, . . . , u. That means that eachpath Ps = ps

1ps2 · · · ps

lsis a path from Cas

to Cbs, where nk−1 ≤ bs < as−1 < as.

We now need the following two claims.

Claim (i). Let R = r1r2 · · · rm′ be a (Cj′ , Ci′)F -path in D, with 1 ≤ i′ < j′ ≤ t. Then any(Cg, Cf )F -path, Q = q1q2 · · · qs in D, with i′ ≤ f < g ≤ j′ has {q+

1 , q−s } ⊆ V c(r+

1 ) ∩ X .

Claim (ii). Let Rk = rk1rk

2 · · · rkmk

be a (Cjk, Cik)F -path in D, for k = 1, 2, with 1 ≤ i1 <

i2 < j1 < j2 ≤ t. Then V c(r1+1 ) = V c(r2+

1 ).

Proof of Claim (i). By Theorem 3.6 we have that {r+1 , r−

m′} ⊆ V c(r+1 ) ∩ X . If (i) is false,

then we can choose a (Cg, Cf )F -path, Q = q1q2 · · · qs, in D, with maximal g (i′ ≤ f < g ≤ j′),such that {q+

1 , q−s } /⊆ V c(r+

1 ) ∩ X . By Theorem 3.6 we have that {q+1 , q−

s } ⊆ V c(q+1 ) ∩ X , so

clearly V c(q+1 ) /= V c(r+

1 ).If f > i′ then let w ∈ V (Cf ) be arbitrary, with r−

m′ → w, and let P p = Ci′ [rm′ , r−m′ ]w. If

f = i′ then let P p = ∅. If g < j′ then from the maximality of g we get that q+1 → r+

1 , andlet P s = q+

1 Cj′ [r+1 , r1], otherwise g = j′ and let P s = ∅. We now define the (Cg, Cf )◦

F -pathS = P sRP p = s1s2 · · · sn, which has V c(s+

1 ) /= V c(q+1 ). This is a contradiction against

Theorem 3.6, since V c∗ (f, g) /= V c(s+

1 ) or V c∗ (f, g) /= V c(q+

1 ).

Proof of Claim (ii). Assume that V c(r1+1 ) /= V c(r2+

1 ). If R1 and R2 are not disjointthen, let x be the first vertex on R1 which lies in V (R1 ∩ R2). Now define the path R =R1[r1

1, x]R2[x, r2m2

] = r1r2 · · · rm′ . By Theorem 3.6 we get V c(r+1 ) = V c(r1+

1 ) /= V c(r2+1 ) =

V c(r2−m2

) = V c(rm′), a contradiction. Therefore we may assume that R1 and R2 are disjoint.If r2−

m2→ r1+

1 , then we get the (Cj2 , Ci1)◦F -path, Q = R2Ci2 [r

2m2

, r2−m2

]Cj1 [r1+1 , r1

1]R1 =q1q2 · · · qs in D, with V c(q+

1 ) = V c(r2+1 ) /= V c(r1+

1 ) = V c(r1−m1

) = V c(q−s ), which is a

contradiction against Theorem 3.6. If r1+1 → r2−

m2then by Claim (i) we get that r1++

1 must be inthe color class V c(r1+

1 ), which is impossible.

Let Vqk= V c(p1+

1 ). Define the statement α(h), to be true if and only if all (Cj , Ci)F -paths,P = p1p2 · · · pl, with nk−1 ≤ i < j ≤ ah, have {p+

1 , p−l } ⊆ Vqk

∩ X . If we can show that α(u)is true then we are done. By Claim (i) we have that α(1) holds. We will use induction, so assumethat α(h) is true for all 1 ≤ h < H ≤ u.

If V c(pH+1 ) /= Vqk

, then, since by the induction hypothesis we have that p(H−1)+1 ∈ Vqk

,Claim (i) (if bH ≤ bH−1) or Claim (ii) (if bH > bH−1) gives us a contradiction. Therefore wemay assume that V c(pH+

1 ) = Vqk. Let P = p1p2 · · · pl be any (Cj , Ci)F -path, with nk−1 ≤ i <

j ≤ aH . If j ≤ aH−1 we are done by the induction hypothesis, and if i ≥ bH then we are doneby Claim (i), so we may assume that nk−1 ≤ i < bH < aH−1 < j ≤ aH . If V c(p+

1 ) /= Vqk,

then the paths PH and P give us a contradiction, by Claim (ii) (if j < aH ) or by Claim (i) (ifj = aH ). Therefore we must have V c(p+

1 ) = Vqk, which together with Theorem 3.6 implies that

{p+1 , p−

l } ⊆ Vqk∩ X . This shows that α(H) is true.

From now on assume that the integers 1 = n0 < n1 < · · · < nm = t and q1, . . . , qm ∈{1, 2, . . . , c} are defined as in Theorem 3.8.

Theorem 3.9. If j /∈ {n0, . . . , nm} and ni−1 < j < ni (1 ≤ i ≤ m), then |Vqi ∩ V (Cj)| =|V (Cj)|/2.


Proof. Let j /∈ {n0, . . . , nm}be arbitrary. Let i (1 ≤ i ≤ m)be chosen such thatni−1 < j <ni. Assume that |Vqi

∩ V (Cj)| /= |V (Cj)|/2. Then there is an x ∈ V (Cj) with {x, x+} ∩ Vqi=

∅. Since j /∈ {n0, . . . , nm} there is a (Cj+1 ∪ · · · ∪ Cni, Cni−1 ∪ · · · ∪ Cj−1)F -path,

P = p1p2 · · · pl, in D. Let p1 ∈ Cj′ and pl ∈ Ci′ , with ni−1 ≤ i′ < j < j′ ≤ ni. Now the cycleC∗ = PCi′ [pl, p

−l ]Cj [x+, x]Cj′ [p+

1 , p1] is a contradiction against the minimality of t.

Theorem 3.10. If j ∈ {n1, . . . , nm−1} then for all x ∈ V (Cj) we have that if x /⇒ V (Cj+1) ∪· · · ∪ V (Ct), then V (C1) ∪ · · · ∪ V (Cj−1) ⇒ x−.

Proof. Let j ∈ {n1, . . . , nm−1} be arbitrary. Assume that the theorem is false, which impliesthat there is an x ∈ V (Cj), with x /⇒ V (Cj+1) ∪ · · · ∪ V (Ct), and V (C1) ∪ · · · ∪ V (Cj−1) /⇒x−. Let z ∈ V (Cj′)(j + 1 ≤ j′ ≤ t) be chosen so z → x, and let y ∈ V (Ci′) (1 ≤ i′ ≤ j − 1)be chosen so that x− → y. We observe that V c(y−) = V c(x) /= V c(x−) = V c(z+), and sincej ∈ {n1, . . . , nm−1} we get that y− → z+. Now the cycle C∗ = Ci′ [y, y−]Cj′ [z+, z]Cj [x, x−]yis a contradiction against the minimality of t.

Theorem 3.11. If P = p1p2 · · · pl is a (Cj , Ci)◦F -path, and R = r1r2 · · · rm is a (Ci, Cj)◦

F -path, such that P and R are disjoint, then either p+

1 /= rm or p−l /= r1 (or both).

Proof. If p+1 = rm and p−

l = r1 then the cycle Ci[rm, p1]PCj [pl, r1]R, is a contradictionagainst the minimality of t.

We summarize the results obtained so far in the following theorem.

Theorem 3.12. Let D be a SMD with color classes V1, V2, . . . , Vc. Let X ⊆ V (D) and let F bea 1-diregular subgraph of D consisting of t cycles, that covers X , such that t is minimum, thenthe following holds.

(a) We can label the cycles C1, C2, . . . , Ct of F , such that Ci '> Cj , whenever 1 ≤ i < j ≤ t.(b) Assume thatC1, C2, . . . , Ct are ordered as stated in (a), then there are cycles, Cn0 , Cn1 , . . . ,

Cnm(n0 = 1, nm = t), and integers q1, q2, . . . , qm ∈ {1, 2, . . . , c}, such that the follow-

ing is true. For all (Cj , Ci)-paths, P = p1 · · · p|P |, with V (P ) ∩ V (F ) = {p1, pl} and1 ≤ i < j ≤ t, there exists an integer k ∈ {1, 2, . . . , m}, such that nk−1 ≤ i < j ≤ nk

and {p+1 , p−

l } ⊆ Vqk∩ X .

(c) Moreover the following holds.(i) If j /∈ {n0, . . . , nm} and ni−1 < j < ni (1 ≤ i ≤ m), then |Vqi

∩ V (Cj)|= |V (Cj)|/2.

(ii) If j ∈ {n1, . . . , nm−1} then for all x ∈ V (Cj) we have that if x /⇒ V (Cj+1) ∪ · · · ∪V (Ct), then V (C1) ∪ · · · ∪ V (Cj−1) ⇒ x−.

(iii) If P = p1p2 · · · pl is a (Cj , Ci)◦F -path, and R = r1r2 · · · rn is a (Ci, Cj)◦

F -path, suchthat P and R are disjoint, then either p+

1 /= rn or p−l /= r1.

By Theorem 4.4 in [3] (see also [7]), and by examining the proofs of Lemma 3.4 and Theorem3.5–3.11, one can obtain the following.

Theorem 3.13. Let D be a SMD, and X ⊆ V (D) be arbitrary. Let F be a 1-diregular subgraphof D, that covers X . Then in O(|V |3) time we can f ind a new 1-diregular subgraph, F ′, of D,that covers X , such that F ′ has the properties (a), (b) and (c) given in Theorem 3.12.

Furthermore we can f ind F ′, such that for all cycles C in F , the vertices X ∩ V (C) areincluded in some cycle of F ′.


FIGURE 1. A non-Hamiltonian bk/2c-strong SMD, with a factor, where |A1| = |B| = bk/2c and|A2| = |C| = dk/2e.

4. CONSEQUENCES OF THEOREM 3.12

A set, I , of vertices in a digraph D is called independent if there are no arcs between the verticesof I . The size of the maximum independent subset of a set, X , is denoted by α(X) and calledthe independence number of X . The following theorem is proved in [1] (as Theorem 3.3 [1]).

Theorem 4.1. If D is a k-strong digraph, and X ⊆ V (D), has independence number α(X) ≤ k,then there is a 1-diregular subgraph, in D, that covers X .

Theorem 4.2. Let D be a (bk/2c + 1)-strong SMD, and let X be an arbitrary set of vertices inD such that X includes at most k vertices from each color class of D. If there is a 1-diregularsubgraph F = C1 ∪· · ·∪Ct, which covers X , then there is a cycle C in D, such that X ⊆ V (C).

Proof. We may clearly assume that F has the properties described in Theorem 3.12, andt ≥ 2, since otherwise we are done. Let {Cn0 , Cn1 , . . . , Cnm} and {q1, q2, . . . , qm} be definedas in Theorem 3.12. Since X contains at most k vertices from each color class we have thatmin{|Vq1 ∩ V (C1) ∩ X|, |Vq1 ∩ V (Cn1) ∩ X|} = r ≤ bk/2c. Assume w.l.o.g. that |Vq1 ∩V (Cn1) ∩ X| = r. Since D is (bk/2c + 1)-strong we get that there exists a (V (Cn1) − (Vq1 ∩V (Cn1) ∩ X)−, V (C1) ∪ · · · ∪ V (Cn1−1))-path in D − (Vq1 ∩ V (Cn1) ∩ X)−, P = p1 · · · pl.Assume that pl ∈ V (Ci)(1 ≤ i < n1). By Theorem 3.12, the (Cn1 , Ci)-path P , which is also a(Cn1 , Ci)F -path, contradicts the minimality of F , since n0 ≤ i < n1 and p+

1 /∈ X ∩ Vq1 .

We cannot weaken the value (bk/2c + 1) of strong connectivity in Theorem 4.2. To seethis consider the SMD D in Figure 1, with V (D) = A1 ∪ A2 ∪ B ∪ C, and color classes{(A1 ∪ A2), B, C}. For any sets X and Y, X ⇔ Y (X ⇒ Y ) means that for all x ∈ X andy ∈ Y we have both x → y and y → x (only x → y). Figure 1, now gives the arc set of D,and the sizes of A1, A2, B and C. Clearly D is bk/2c-strong, and X = V (D) contains at mostk vertices from each color-class. D contains a factor, since D〈A1 ∪ B〉 and D〈A2 ∪ C〉 containHamilton cycles. If D has a Hamilton cycle, then every second vertex in the Hamilton cycle mustbe a vertex from A1 ∪ A2, since |A1 ∪ A2| = |V (D)|/2. But the only way to come from A2 ∪ Cto A1 ∪ B, is through an arc c → b, where c ∈ C and b ∈ B, so a Hamilton cycle must includesuch an arc. This shows that the (bk/2c)-strong SMD D has a factor but no Hamilton cycle.

Consider the semicomplete bipartite digraph H , shown in Figure 2, for an example that thebound for strong connectivity in Theorem 4.2 is also best possible when we only consider semi-complete bipartite digraphs. Figure 2 gives the arc set of H , and the sizes of A1, A2, B1, B2 andB3. Let X = B1 ∪ B2, and observe that there is a 1-diregular subgraph, F = C1 ∪ C2 of H ,which covers X , with A1 ∪ B1 = V (C1) and A2 ∪ B2 = V (C2). Assume that there is a cycle,C, in H with X ⊆ V (C). Since |A1 ∪ A2| = |X| every second vertex of the cycle C must comefrom A1 ∪ A2, which implies that V (C) = A1 ∪ A2 ∪ X = A1 ∪ A2 ∪ B1 ∪ B2. However we


FIGURE 2. bk/2c-strong semicomplete bipartite digraph, with a 1-diregular subgraph, but withouta cycle covering B1 ∪ B2, where |B1| = |B3| = |A1| = bk/2c and |B2| = |A2| = dk/2e.

need a vertex from B3 to come from B1 ∪ A1 to B2 ∪ A2. This shows that there is no cycle, C,in H with X ⊆ V (C), even though D is bk/2c-strong.

Corollary 4.3. Let D be a k-strong SMD, and let X ⊆ V (D) be an arbitrary set of vertices inD, with at most k vertices from each color class. Then there exists a cycle C, with X ⊆ V (C).

Proof. From Theorem 4.1 we get that there is a 1-diregular subgraph in D covering X . NowTheorem 4.2 implies this corollary.

Corollary 4.3 immediately implies the following two corollaries.

Corollary 4.4. If a k-strong SMD D has at most k vertices in each color-class, then D containsa Hamilton cycle.

Corollary 4.5. A k-strong SMD has a cycle through any set of k vertices.

We observe that Corollary 4.4 and Corollary 4.5 give affirmative answers to the conjecture byGuo and Volkmann as well as to the conjecture by Bang-Jensen, Gutin and Yeo, described in theintroduction.

Definition 4.6. Let R be a digraph, and let X, Y ⊆ V (R) be arbitrary. Let dR(X, Y ) = |{xy ∈A(R)|x ∈ X, y ∈ Y }|.Theorem 4.7. [9] If R is a diregular digraph, then R contains a factor.

Theorem 4.8. Every diregular SMD is Hamiltonian.

Proof. Let R be a diregular SMD. By Theorem 4.7, R has a factor F = C1 ∪ C2 ∪ · · · ∪ Ct.We may assume that F is chosen, such that t is minimum. If t = 1 then R is Hamiltonian, soassume that t > 1.

Let X = V (R). Let {Cn0 , Cn1 , . . . , Cnm} and {q1, q2, . . . , qm} be defined as in Theorem3.12. Let yx ∈ A(R) be an arc from y ∈ V (Ci), with i ∈ {2, 3, . . . , t} to x ∈ V (C1). Part (b)of Theorem 3.12 implies that x−, y+ ∈ Vq1 . Now we define the two distinct arcs a1(yx) = xy+

and a2(yx) = x−y. By Theorem 3.12, a1(yx) and a2(yx) are arcs in R.If y′x′ and yx are distinct arcs from V (R) − V (C1) to V (C1), then we see that a1(yx), a2(yx),

a1(y′x′) and a2(y′x′) are four distinct arcs from V (C1) to V (R) − V (C1).We have now shown that the number of arcs leaving V (C1) is at least double as large as the

number of arcs entering V (C1). However this contradicts the fact that R is an eulerian digraph.(Since R is diregular, R is eulerian and dR(X, V (R) − X) = dR(V (R) − X, X).)


We observe that Theorem 4.8 gives an affirmative answer to the conjecture by C.-Q. Zhang,which was mentioned in the introduction.

5. MINIMAL FACTORS

The following theorem shows that the conditions of Theorem 3.12 are quite strong.

Theorem 5.1. Let D be a strong SMD, with color classes, V1, . . . , Vc, and let F = C1 ∪ C2 ∪· · · ∪ Ct be a factor of D, with Ci '> Cj , for all 1 ≤ i < j ≤ t. Let {Cn0 , Cn1 , . . . , Cnm}(n0 = 1 and nm = t) be all the cycles in F , such that D − V (Cnk

) is not strong, for k =1, 2, . . . , m − 1. Assume that 1 = n0 < n1 < · · · < nm = t and let Q be all the internal arcs ofthe cycles {Cn0 , Cn1 , . . . , Cnm

}. Now def ine the following three statements.

(a) There exist integers q1, q2, . . . , qm, such that all (Cj , Ci)F -paths, P = p1p2 · · · pl, withnk−1 ≤ i < j ≤ nk have {p+

1 , p−l } ⊆ Vqk

, for all i = 1, 2, . . . , m.(b) If j /∈ {n0, . . . , nm} and ni−1 < j < ni (1 ≤ i ≤ m), then |Vqi

∩ V (Cj)| = |V (Cj)|/2.(c) If j ∈ {n1, . . . , nm−1} then for all x ∈ V (Cj) we have that if x /⇒ V (Cj+1) ∪ · · · ∪

V (Ct), then V (C1) ∪ · · · ∪ V (Cj−1) ⇒ x−.

If (a), (b) and (c) are true, then F is a minimal factor of D − Q.

Proof. Assume that (a), (b) and (c) hold, but there is another factor F ′ = C ′1 ∪ · · · ∪ C ′

s inD − Q, with 1 ≤ s < t.

Since s < t, the factor F ′ must contain some arcs, which join different cycles in F . Further-more, F ′ must contain arcs which go from a cycle in F to a cycle with lower index in F . Wemay assume that e′ = y′x′ is such an arc from Cj to Cj′ (j > j′) in F , such that j is maximum.From the definition of j, we see that F ′ has no arcs that join a cycle of F with another cycle ofF , which has an index larger than j. Indeed if F ′ contains an arc going out of a cycle Cp in F ,then F ′ must also contain an arc going into Cp.

Let i be defined, such that ni−1 < j ≤ ni. Let k be the largest integer (ni−1 ≤ k ≤ j),such that there is a vertex x ∈ V (Ck), that is not in Vqi

, and a y ∈ (V (Cni−1) ∪ · · · ∪V (Cj)) − V (Ck), such that the arc e = yx is in F ′. Such an arc exists, since the arc e′ = y′x′

has y′ ∈ (V (Cni−1) ∪ · · · ∪ V (Cj)) − V (Cj′) and x′ ∈ V (Cj′) − Vqi .Consider the following three cases, which exhaust all possibilities.

ni−1 < k < ni: We note that if a cycle in F ′ enters Ck at some vertex then it must leave Ck again.The path between entering and leaving Ck gives rise to a path in Ck. Using this procedurewe divide Ck up into a number of disjoint paths (possibly of length 0), where one of the pathsstart at x. Since x /∈ Vqi

and |Vqi∩ V (Ck)| = |V (Ck)|/2, one of the paths of Ck must end in

a vertex w, which is in Vqi. Clearly the succesor of w in F ′ cannot be in a cycle of F , with

smaller index than k, since w+ /∈ Vqi. The succesor of w in F ′ cannot be in a cycle of F , with

larger index either, because of the maximality of k. This shows that the succesor of w in F ′

can't lie anywhere, which is a contradiction.k = ni: The successor of x− in F ′ cannot be in a cycle of F with larger index than k, because of

the maximality of j and k = j (because k ≤ j ≤ ni = k). The succesor of x− in F ′ cannotbe in a cycle of F with smaller index either since (x−)+ /∈ Vqi

. Finally the succesor of x− inF ′ cannot belong to Ck, since we have deleted the internal arcs of Ck. This is a contradiction.

k = ni−1: From the definition of e = yx, we see that y lies in a cycle of F , with higher indexthan k, which implies that {y+, x−} ∈ Vqi . The successor of x− in F ′ cannot belong to


a cycle of F with smaller index, since (c) implies that V (C1) ∪ · · · ∪ V (Cni−1−1) ⇒x−. The succesor of x− in F ′ cannot be in a cycle of F with larger index than k, becauseof the maximality of k (and x− ∈ Vqi

). Analog to the case when k = ni, x− cannot be

in Cni−1 .

Theorem 3.12 describes which properties the arcs between different cycles in a minimal factormust possess. By Theorem 5.1, if the arcs between different cycles in a factor has these propertiesand the factor is not minimal, then each minimal factors must use some internal arcs of somespecified cycles.

6. ACKNOWLEDGMENTS

I would like to thank my supervisor Jørgen Bang-Jensen, for introducing me to the problem, andhim and Gregory Gutin for many fruitful discussions and helpful remarks concerning this paper.

References

[1] J. Bang-Jensen, G. Gutin, and A. Yeo, On k-strong and k-cyclic digraphs, Discrete Math. to appear.

[2] J. Bang-Jensen and Y. Manoussakis, Cycles through k vertices in bipartite tournaments, Combinatorica14 (1994), 243–246.

[3] J. Bang-Jensen, G. Gutin, and J. Huang, A sufficient condition for a complete multipartite digraph tobe hamiltonian, Discrete Math. to appear.

[4] J. Bang-Jensen, G. Gutin, and H. Li, Sufficient conditions for a digraph to be hamiltonian, J. GraphTheory 22 181–187 (1996).

[5] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, MacMillan Press, New York (1976).

[6] G. Gutin, Cycles and paths in semicomplete multipartite digraphs, theorems and algorithms: A survey,J. Graph Theory 19 (1995), 481–505.

[7] G. Gutin, Paths and Cycles in Directed Graphs. Ph.D. thesis, Tel Aviv (1993).

[8] A. J. Hoffmann, Some recent applications of the theory of linear inequalities to extremal combinatorialanalysis, Proc. Symp. Appl. Math. 10 (1960), 113–127.

[9] O. Ore, Theory of Graphs, AMS Colloquium Publications 38, AMS, Providence (1962).

[10] L. Volkmann, a talk at The Second Krakow Conference of Graph Theory (1994).

[11] C.-Q. Zhang, Hamilton paths in multipartite oriented graphs. Ann. Discrete Math. 41 (1989), 499–514.

Received July 5, 1996

One-diregular subgraphs in semicomplete multipartite digraphs

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