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 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) 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, 2k  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 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 graphs Ming Tsai, Tsung-Han Tsai, Jimmy J. M On 4-ordered 3-regular graphs Ming Tsai, Tsung-Han Tsai, Jimmy J.M. Tan, Lih-Hsing Hsu Mathematical 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 graphs LH Hsu, JIMMY J. M On an open problem of 4-ordered hamiltonian graphs LH Hsu, JIMMY J. M. Tan, E Cheng, L Liptak, C.K. a M. Tsai submitted 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 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) 1 3 2 4

1 3 2 4

1 3 2 4

1 3 2 4

1 3 2 4

1 3 2 4

1 3 2 4 1,3 no 2,4

1 3 2 4

1 3 2 4 3,4 no 1,2

1 3 2 4 3,4,2

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.

1 3 2 4

1 3 2 4

Case 1 1 3 2 4

1 3 2 4

Case 2 1 3 2 4

Case 3 1 3 2 4

1 3 2 4

Step 1: Case 1 is when the first column is the (3q)th column counting counterclockwise from the column containing 1 for some q. 1 3 2 4 N

Case 2 is when the first column is the (3q + 2)nd column counting counterclockwise from the column containing 1 for some q. 1 3 2 4 N 1’ 2’

Case 3 is when the first column is the (3q + 1)st column counting counterclockwise from the column containing 1 for some q. 3 2 4 1’ N 2’

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. 1” 1’ 2’ N 2”

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. 1’ 2’ 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. 1” 1’ 2’ N 2”

Step 3 1 3 2 4

Case 1 3 2 4 1’ 2’

Case 2 3 2 4 1’ 2’

Case 3 3 2 4 t 2’ 1’

Lemma 3. 5. Suppose n is even and 28 ≤ n ≤ 50 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 6

Case 2 6

Case 3 6

Case 4 6

Case 5 6

Case 6 4

Case 7 6

Case 8 6

Case 7 Case 1 Case 9 Case 10 Case 8 Case 6 Case 11 Case 12

Case 13 12

Case 14 12

Case 15 12

Case 16 12

n  50 computer result n52 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 n88. 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 n12.

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 hypercubes C. K. Lin, Jimmy J. M The ordered bipancyclic properties of hypercubes C.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