Disjoint Hamiltonian cycles in graphs

Let G be a 2(k + I)-connected graph of order n. It is proved that if uv rJ. E(G) implies that max{d(u), d(v)} ?: ~ + 2k then G contains k + 1 pairwise disjoint Hamiltonian cycles when c5 (G) ?: 4k + 3. 1 .. Introduction All graphs we consider are finite and simple. We use standard terminology and notation from Bondy and Murty [2] except as indicated. Let G = (V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For a subset U of V(G), G[U] is the subgraph of G induced by U. For two disjoint subsets (resp. subgraphs) S, T of V(G) (resp. G), put E(S, T) = {st E E(G)I s E S, t E T}, E(S, T) = {st rJ. E(G)I s E S, t E T}, NT(S) = {t E TI t is adjacent to some vertex in S}, NT(S) = {t E TI t is not adjacent to any vertex in S}, dT(S) = INT(S)I, dT(S) = INT(S)I; when S = {s}, we write dT(s) and dT(s) for dT( {s}) and dT( {s}). Let P = uv·· . w and Q = xy· .. z be two vertex-disjoint paths of G. If ux and wz are the edges of G, we denote by PQ the cycle P U Q U {ux, wz} with a given orientation in the order from x to z along the path Q. For a cycle C with an given orientation and a vertex v E V (C), we denote by va and vi: the predecessor and successor of v on C, respectively. Two Hamiltonian cycles are called disjoint when they share no common edge, and similar terminology will be applied to disjoint paths. "'This research was supported by CityU Strategic Research Grant 7000743, and by the National Natural Science Foundation of China and by the Doctoral Discipline Foundation. Australasian Journal of Combinatorics 19(1999), pp.S3-S9 Let x be a real number. We denote by [x] the maximum integer less than or equal to x. The following theorem due to Geng-hua Fan [3] is well known. Theorem A If a 2-connected graph G of order n satisfies the condition d( u, v) = 2 =* max{d(u), d(v)} ~ %, then G contains a Hamiltonian cycle. The proof of Fan's result was simplified by F.Tian [5]. In 1993, S.Zhou [6] proved the following theorem by using the method essentially same as used by Tian. Theorem B If a 4-connected graph G of order n satifies the condition d( u, v) = 2 =* max{d(u),d(v)} ~ % +2, then G contains 2 Hamiltonian cycles. On disjoint Hamiltonian cycles, H.Li [4] proved in 1989 the following interesting result. Theorem C Let nand k be positive integers such that n ~ 8k 5, and let G be a graph on n vertices with minimum degree 8 satisfying 2k + 1 :::; 8 :::; 2k + 2. If da ( u) + da (v) ~ n for any pair of nonadjacent vertices u and v, and if h, ... , lk are integers satisfying 3 :::; it :::; l2 ... :::; lk :::; n, then G contains k disjoint cycles of length h, l2,' . " lk' respectively. In particular, under these conditions G contains k disjoint Hamiltonian cycles. There are many results on Hamiltonian cycles, but few on disjoint Hamiltonian cycles. Here we focus our attention on the study of disjoint Hamiltonian cycles in graphs. As in [6], for a nonnegative integer k, a graph of order n is called a Fan 2k-type graph if d(u, v) = 2 implies max{d(u) , d(v)} ~ % +2k. In this paper, we call a graph of order n an Ore 2k-type graph if uv ¢ E(G) implies max{d(u), d(v)} ~ % + 2k. We will prove the following Theorem. Theorem 1 Let G be a 2(k + I)-connected Ore 2k-type graph. If 8(G) ~ 4k + 3 then G contains k + 1 disjoint Hamiltonian cycles. We surmise the condition 8(G) ~ 4k+3 can be deleted, but this task is formidable. So we pose the following conjecture. Conjecture 1 For any nonnegative integer k, every 2(k + I)-connected Ore 2ktype graph contains k + 1 disjoint Hamiltonian cycles. It is easy to see that the proof of Conjecture 1 will be a stepping stone in the proof of the following conjecture 2 posed by S.Zhou in [6]. Conjecture 2 For any nonnegative integer k, every 2(k + I)-connected Fan 2ktype graph contains k + 1 disjoint Hamiltonian cycles. We will prove Theorem 1 in section 2. As an application of the method established in section 2, we will give, in section 3, an alternative proof of Conjecture 1 for k = 1. We attempt to explain how the method established in section 2 might be useful in proving the conjecture 1. 2. Proof of Theorem 1 In this section, all graphs we consider are 2(k + l)-connected Ore 2k-type. The