A structural method for Hamiltonian graphs

In this paper, we shall introduce a special structure for graphs and show that a graph G is hamiltonian if and only if G has such a special structure. Using this result, we can prove a new weakened version of Fan's condition for hamiltonian graphs, which a recent result of Bedrossian, Chen and Schelp (1993). 1 Preliminaries and Main Results We consider only finite undirected graphs without loops or multiple edges. The set of vertices of G is denoted by V(G) or just by V; the set of edges by E(G) or just by E. We lise IGI a symbol for the cardinality of V(G). If Hand are subsets of V(G) or subgraphs of G, we denote by NH(S) the set of vertices in H which are adjacent to some vertex in S, and set dH(S) = IN H(S)I. If S' = {u} and H G, then let NG(u) = N(u) and set dG(u) = d(u). For D ~ V(G), OlD) denotes the s~bgraph of G induced by D. For basic graph-theoretic terminology, we refer the reader to [3]. Definition 1. Let H be a subgraph of G and x, y E V(G) \ V(H). {x, y} is called a pair of useful vertices of H if G[H U {x, V}] contains a hamiltonian path connecting x and y. Definition 2. A graph G is call L-decomposable if G can be separated into k + 1 pairwise disjoint s)lbgraphs Go, G 1 , ... , Gk such that the following four conditions are satisfied: 1) Go is complete. 2) For any 1 ~ i ~ "-, there exists a subset Si ~ NGo(Gd with at least two vertices which contains a vertex x such that for every y (:;;f x) E Si, {::e,y} is a pair of useful vertices of Gi. 3) For any three distinct Sj, Sj, SI, we have S'i n Sj n SI = 0. 1 Supported by National Education Committee of China and Chinese Academy of Sciences Australasian Journal of Combinatorics !l( 1995 L pp.257-262 4) For any integer r k, I U1:Sj:Sr 1= r if and only if IV(Go)1 k r. If G then we say the partition GO,G 1 ,Gk which satisfies the four conditions above a of In Section 2, we shall prove the following structural theorem. Theorem 1. A graph G is hamiltonian if and only if G has a Theorem 1 bas some CLPP"<--CLv'VHC:>. We shall some ,-,-'.,,0,'"1-'1','" here. In order to do this, we need some additional' terminology and notations. In 1, WI'" define four kinds of F-graph. and N-graph. v y