A clique-covering sufficient condition for hamiltonicity of graphs

Abstract

International audienceA graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as NP-complete problem and no satisfactory characterization for hamiltonian graphs has been found. There are several necessary conditions for hamiltonicity and since the seminal work of Dirac in 1952, many sufficient conditions were found. These conditions are usually expressed in terms of node degree, connectivity, density, toughness, independent sets, regularity and forbidden subgraphs. In this article we give an extended clique decomposition condition ensuring the hamiltonicity of a large class of graphs. Then we discuss briefly the possibility of broader extensions as well as algorithmic issues

Normal eulerian clique-covering and hamiltonicity

“…This condition is a generalization of a result from [6]. The main result of [1] states that if there exists an eulerian clique-covering of a graph G then the graph is hamiltonian. A clique-covering is eulerian if it is pairwise-joint and evenly-joint (cf.…”

“…In [1], a sufficient condition for hamiltonicity of graphs based on clique-coverings is introduced. This condition is a generalization of a result from [6].…”

“…Notice that a clique is here a subset of vertices which induces a complete subgraph of G but not necessarily a maximal one. Notice also that a clique-covering in the sense of [1] covers both the vertices and the edges of the graphs (and not only the vertices, as defined for instance in [5]). E-mail address: vallee_th@yahoo.fr.…”

“…In this article, we first recall some basic definitions and results from [1]. Then we prove several new results concerning pairwise-joint and normal pairwise-joint cliquecoverings.…”

“…As a consequence, in order to decide the existence of an eulerian clique-covering of a graph, it is sufficient to decide the existence of a normal one. Hence the algorithm given in [2] is fully general relative to the sufficient condition for hamiltonicity given in [1].…”