Normal eulerian clique-covering and hamiltonicity

Abstract

A connected graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as an NP-complete problem and no satisfactory characterization for these graphs has been found. Since the seminal work of Dirac in 1952 many sufficient conditions were found. In 1974, Goodman and Hedetniemi gave such a condition based on the existence of a clique-covering of the graph. This condition was recently generalized using the notion of eulerian clique-covering. In addition, an algorithm able to find a normal eulerian clique-covering for a large class of graphs was also introduced. A normal clique-covering has additional properties, making the search for such a covering easier than in the general case. In this article, we prove several properties of normal clique-coverings. In particular we prove that there exists an eulerian clique-covering of a graph if and only if there exists a normal one. Using this result, the search for an eulerian clique-covering as a sufficient condition for hamiltonicity can be reduced to the normal case.