A study of k-ordered hamiltonian graphs

Story begins

L. Ng, M. Schultz, k-Ordered Hamiltonian graphs, J. Graph Theory 24 (1997) 45–57.

• Let G be a hamiltonian graph of order n. For a positive integer k with k n, we say that G is k-ordered ((hamiltonian)hamiltonian) if for every sequence S: v1, v2,…,vk of k distinct vertices, there exists a hamiltonian cycle C of G such that the vertices of S are encountered on C in the specified order.

• Proposition 1. Let G be a hamiltonian graph of order n 3. If G is k-ordered, 3 k n, then G is (k − 1)-connected.

• Corollary 2. If G is a k-ordered hamiltonian graph, then (G) k − 1.

• Theorem A. Let G be a graph of order n 3 and let k be an integer with 3 k n.

If deg u + deg v n + 2k −6 for every pair u, v of nonadjacent vertices of G, then G is a k-ordered hamiltonian graph.

• We define a graph G of order n 3 to be k-ordered hamiltonian-connected, or more simply k-hamiltonian-connected, 2k n, if for every sequence v1, v2,…,vk of k distinct vertices, G contains a v1-vk hamiltonian path that encounters the vertices v1, v2,…,vk in this order.

• Theorem B. Let G be a graph of order n 4 and let k with 4 k n be an integer.

If deg u+ deg v n + 2k − 6 for every pair u, v of nonadjacent vertices of G, then G is k-hamiltonian-connected.

• Problem 1. Determine the best possible degree condition for Theorem A.

• Problem 2. Determine whether there is an infinite class of 3-regular 4-ordered graphs.

• Problem 3. Determine the best possible degree condition for Theorem B.

• Problem 4. Study the existence of small degree k-hamiltonian-connected graphs.

Our First Choice

• Problem 2. Determine whether there is an infinite class of 3-regular 4-ordered graphs.

K. Meszaros, On 3-regular 4-ordered graphs, Discrete Math. 308 (2008) 2149–2155.

• A simple graph G is k-ordered if for any sequence of k distinct vertices v1, v2,…,vk of G there exists a cycle in G containing these k vertices in the specified order.

• In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K4 and K3,3.

• 3-regular 4-ordered

• Petersen graph

• Generalized Honeycomb torus GHT(3, n, 1) for n is an even integer with n ≥ 8.

• 3-regular 4-ordered Hamiltonian graph

• Heawood graph

Petersen Graph

GHT(3,12,1)

Heawood graph

• An Infinite Contruction of 3Regular 4Ordered Graphs

• CHENG Kun, A Yongga (College of Mathematics Science,Inner Mongolia Normal University,Huhhot 010022,China)  

On 4-ordered 3-regular graphsMing Tsai, Tsung-Han Tsai, Jimmy J.M. Tan, Lih-Hsing HsuMathematical and Computer Modelling, Vol. 54, (2011), 1613—1619.

• Theorem. Assume that m is an odd integer with m ≥ 3 and n is an even integer with n ≥ 4. The generalized honeycomb torus GHT(m, n, 1) is 4-ordered if and only if n 4.

• Theorem. Assume that m is a positive even integer with m ≥ 2 and n is an even integer with n ≥ 4. The generalized honeycomb torus GHT(m, n, 0) is 4-ordered if and only if n 4.

GHT(4,12,0)

4-ordered 3-regular cells

On an open problem of 4-ordered hamiltonian graphsLH Hsu, JIMMY J. M. Tan, E Cheng, L Liptak, C.K. a M. Tsaisubmitted

P(8,1) P(8,2) P(8,3)

• Proposition. [9] No 4-ordered 3-regular graph with more than six vertices contains a 4-cycle.

• Lemma. P(n, 1) is neither 4-ordered nor 4-ordered Hamiltonian.

• Theorem. P(5, 2) is 4-ordered but not 4-ordered Hamiltonian. If n>5, then P(n, 2) is not 4-ordered, and hence, not 4-ordered Hamiltonian.

The graph P(n, 2)

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• Theorem. Let n 7 be even. Then P(n, 3) is 4-ordered if and only if n {7, 9, 12}. In addition, P(n, 3) is 4-ordered Hamiltonian if and only if n is even and either n = 18 or n 24.

• We start with the following two results whose validity were checked by a computer.

• Lemma 3.1. Let 7 ≤ n ≤ 26. Then P(n, 3) is not 4-ordered if and only if n {7, 9, 12}∈ .

• Lemma 3.2. Let 7 ≤ n ≤ 26. Then P(n, 3) is 4-ordered Hamiltonian if and only if n {18, 24, 26}∈ .

• Theorem 3.3. Let n ≥ 7. Then P(n, 3) is 4-ordered unless n {7, 9, 12}∈ .

16 cases

Case 1 Case 2

Case 3 Case 4

Case 5 Case 6

Case 7 Case 8

Case 9 Case 10

Case 11 Case 12

Case 13 Case 14

Case 15 Case 16

• Theorem 3.4. Let n ≥ 7 be odd. Then P(n, 3) is not 4-ordered Hamiltonian.

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Step 1: Case 1 is when the first column is the (3q)th column counting counterclockwise from the column containing 1 for some q.

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Case 2 is when the first column is the (3q + 2)nd column counting counterclockwise from the column containing 1 for some q.

Case 3 is when the first column is the (3q + 1)st column counting counterclockwise from the column containing 1 for some q.

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Step 2: We look for the next column to be used going counterclockwise from the dead-end configuration.

Case 1 is presented in Figure, where the first column in C counterclockwise from the left configuration is the (3q + 2)nd column from the column containing 10, and it is marked by N.

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Case 2 is presented in Figure, where the first column in C counterclockwise from the left configuration is the (3q)th column from the column containing 1’, and it is marked by N.

Case 3 is presented in Figure, where the first column in C counterclockwise from the left configuration is the (3q + 1)th column from the column containing 1’, and it is marked by N.

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• Lemma 3.5. Suppose n is even and 28 ≤ n ≤ 50. Then P(n, 3) is 4-ordered Hamiltonian.

• Theorem 3.6. Suppose n ≥ 7 is even. If n = 18 or n ≥ 24, then P(n, 3) is 4-ordered Hamiltonian.

Case 1

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• n 50 computer result

• n52 induction

Free Ticket: Problem 4

• Theorem. If n 7 and n is even, then P(n,3) is 4-ordered Hamiltonian laceable if and only if n 10.

• If n 7 and n is odd, then P(n, 3) is 4-ordered Hamiltonian connected if and only if n = 15 or n 19.

Similar technique• D. Sherman, M. Tsai, C.K. Lin, L. Liptak, E. Chang, J.J.M. Tan, and L.H. Hsu (2010),

"4-ordered Hamiltonicity for Some Chordal Ring Graphs," Journal of Interconnection Networks, Vol. 11 pp. 157-174.

• The Chordal Ring graph, denoted by CR(2n, 1, k), has its second parameter fixed and k odd such that 3 k n. The vertex set is V (CR(2n, 1, k)) = {vi | 0 i < 2n}, while the edges are of two types as follows: E(CR(2n, 1, k)) = {(vi, vi+1) | 0 i < 2n} {(vi, vi+k) | i is even and 0 i < 2n}, where the indices are always taken modulo 2n.

Heawood graph= CR(14,1,5)

• Theorem. Let n 5. Then CR(2n, 1, 5) is 4-ordered Hamiltonian if and only if 2n = 12k + 2 or 2n = 12k + 10 for some k 2 or 2n = 14.

Similar technique?!

• P(n,4)

• 4-ordered hamiltonian and 4-ordered hamiltonian connected

• Need computer result n88.

• So, just do hamiltonian connected. (submitted)

• On the Hamilton connectivity of generalized Petersen graphs Brian Alspach and Jiping Liu, Discrete Mathematics 309 (2009) 54615473

• We investigate the Hamilton connectivity and Hamilton laceability of generalized Petersen graphs whose internal edges have jump 1, 2 or 3.

• P(n,4) is hamiltonian connected iff n12.

Problem 1

• R. Li (2010), "A Fan-type result on k-ordered graphs," Information Processing Letters, Vol. 110, pp. 651-654

• Theorem. Let k 3 be an integer and let G be a graph on n 100k vertices with d(x) + d(y) n for any two vertices x and y with d(x, y) = 2. If G is 3k/2-connected, then G is k-ordered hamiltonian.

Problem 3

• Theorem. Let k 3 be an integer and let G be a graph on n 99k vertices with d(x) + d(y) n+1 for any two vertices x and y with d(x, y) = 2. If G is (3k/2-1)-connected, then G is k-ordered hamiltonian connected.

• (manuscript: Not guarantee)

Problem 1

• Theorem. Let k 3 be an integer and let G be a graph on n 99k vertices with d(x) + d(y) n for any two vertices x and y with d(x, y) = 2. If G is 3k/2-connected, then G is k-ordered hamiltonian.

• (manuscript: Not guarantee)

Discussion

• Computer search is a dirty work for mathematician. However, it give us a possible direction.

• Generalized Petersen graphs• Generalized Chordal ring• Honeycomb rectangular torus• Generalized Honeycomb torus

• Most of them are 4-ordered hamiltonian, 4-ordered hamiltonian laceable/4-orderd hamiltonian connected. We need a neat proof.

New Directions

• Cells

• k-regular (k+1) ordered graphs

The ordered bipancyclic properties of hypercubesC.K. Lin, Jimmy J. M. Tan, C.N. Hung, and L.H. Hsu

• Let x1,x2,x3,x4 be any four vertices in the hypercube Qn with n 5. Let l be any even integer satisfying h(x1, x2) + h(x2,x3) + h(x3,x4) + h(x4,x1) l 2n. We will prove that there exists a cycle C in Qn of length l such that C traverses these 4 vertices in the specific order except for the case that l {6,8} when x1,x3,x2,x4, x1 forms a cycleof length 4.

• Thanks