Forbidden subgraphs and Hamiltonian properties of graphs

We consider only finite undirected graphs without loops or multiple edges. Notation or terms not defined here can be found in [1]. Let G be a graph and let S £; V( G). The subgraph (S) induced by S is the graph with vertex set Sand Whose edge set consists of those edges of G incident with two vertices of S. The distance d(u, v) between vertices u and v in a connected graph G is the minimum number of edges in a u v path. The diameter of a graph G is maxU,VEV(G) d(u, v). A graph is hamiltonian (traceable) if it has a cycle (path) containing all its vertices. A pancyclic graph of order p contains a cycle of length I for each I (3 os;; los;; p). A graph is panconnected if, for each pair u, v of distinct vertices, there is a u v path of length I for each I (d (u, v) E;; IE;; P 1). A graph G is homogeneously traceable, if, for each vertex v in G, there exists a hamiltonian path with initial vertex v. Homogeneously traceable nonhamiltonian graphs exist for all orders p, except 3 os;; p E;; 8 (see [2J). The following implications are well-known and the reverse implications fail to hold: panconnected ~ pancyclic ~ hamiltonian ~ homogeneously traceable.