Supereulerian Graphs and Chvátal-Erdős Type Conditions

Abstract

In 1972, Chvatal and Erdős showed that the graph \(G\) with independence number \(\alpha(G)\) no more than its connectivity \(\kappa(G)\) (i.e., \(\kappa(G) \geq \alpha(G)\)) is hamiltonian. In this paper, we consider a kind of Chvatal and Erdős type condition on edge-connectivity \(\lambda(G)\) and matching number (edge independence number). We show that if \(\lambda(G) \geq \alpha'(G) – 1\), then \(G\) is either supereulerian or in a well-defined family of graphs. Moreover, we weaken the condition \(\kappa(G) \geq \alpha(G) – 1\) in [11] to \(\lambda(G) \geq \alpha(G) – 1\) and obtain a similar characterization on non-supereulerian graphs. We also characterize the graph which contains a dominating closed trail under the assumption \(\lambda(G) \geq \alpha'(G) – 2\).