Sufficient Conditions for a Digraph to be Supereulerian

A (di)graph is supereulerian if it contains a spanning, connected, eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we give a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc-connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasi-transitive and verify the analogous statement for undirected graphs.