Long dominating cycles and paths in graphs with large neighborhood unions

Let G be a graph of order n and define NC(G) = min{l/V(u) U N(u)l Iuu $& E(G)}. A cycle C of G is called a dominating cycle or D-cycle if MG) MC) is an independent set. A D-path is defined analogously. The following result is proved: if G is 2-connected and contains a D-cycle, then G contains a D-cycle of length at least rnin(n, WCIG)} unless G is the Petersen graph. By combining this result with a known sufficient condition for the existence of a D-cycle, a common generalization of Ore's Theorem and several recent "neighborhood union results" is obtained. An analogous result on long D-paths is also established. 1. TERMINOLOGY AND NOTATIONS We use [3] for terminology and notations not defined here, and consider simple graphs only. Throughout, let G be a graph of order n. If G has a Hamilton cycle (a cycle containing every vertex of G), then G is called hamiltonian. G is traceable if G has a Hamilton path (a path containing every vertex of G). A cycle C of G is called a dominating cycle, or briefly D-cycle, if V(G) V(C) is an independent set of vertices in G. A dominatingpath or D-path is analogously defined. Two edges e l and ez of G are called remote if they are nonadjacent, and there is no edge of G joining an end of el and one of e2. The degree of an edge uu of G is the number of vertices in V(G)-{u,u} adjacent to at least one of the vertices u and u. Journal of Graph Theory, Vol. 15, 29-38 (1991)