Recent research in Graph Theory

A well-known and fundamental property of graphs is Hamiltonicity. A connected graph is Hamiltonian if it contains a spanning cycle. Determining if a graph is Hamiltonian is known as a NP-complete problem and no satisfactory characterization exists. Nevertheless, many sufficient conditions for Hamiltonicity were found, usually expressed in terms of degree, connectivity, density, toughness, independent set, regularity and forbidden subgraphs. In [20], Goodman and Hedetniemi gave two alternative sufficient conditions based on the existence of some type of clique-covering of the graph. In [13] one of these conditions is generalized using the notion of eulerian clique-covering. A polynomial algorithm to decide the existence of such a covering for every graph containing at least one simplicial vertex is also given in [12]. Now, several Hamiltonicity preserving closure concepts for claw-free graphs were defined recently in [25, 15]. The closure of the graph is Hamiltonian if and only if the graph is. Moreover, it is easier to look for a clique-covering of the closure than to look for a clique-covering of the graph itself. Starting from these facts, I defined recently three new Hamiltonicity preserving closure concepts for graphs (cf. [14, 29, 30]).