Some Hamiltonian Properties Of One-Conflict Graphs

Dirac's and Ore's conditions (1952 and 1960) are well known and classical sufficient conditions for a graph to contain a Hamiltonian cycle and they are generalized in 1976 by the Bondy-Chvatal Theorem. In this paper, we add constraints, called conflicts. A conflict in a graph G is a pair of distinct edges of G. We denote by (G, Conf) a graph G with a set of conflicts Conf. A path without conflict P in (G, Conf) is a path P in G such that for any edges e, e' of P, {e, e'} is not an element of. Conf. In this paper we consider graph with conflicts such that each vertex is not incident to the edges of more than one conflict. We call such graphs one-conflict graphs. We present sufficient conditions for one-conflict graphs to have a Hamiltonian path or cycle without conflict.