The Chvátal–Erdős Condition for a Graph to Have a Spanning Trail

Abstract

Chvátal and Erdős proved a well-known result that every graph \(G\) with connectivity \(\kappa (G)\) not less than its independence number \(\alpha (G)\) is Hamiltonian. Han et al. (in Discret Math 310:2082–2090, 2003) showed that every 2-connected graph \(G\) with \(\alpha (G)\le \kappa (G)+1\) is supereulerian with some exceptional graphs. In this paper, we investigate the similar conditions and show that every 2-connected graph \(G\) with \(\alpha (G)\le \kappa (G)+3\) has a spanning trail. We also show that every connected graph \(G\) with \(\alpha (G)\le \kappa (G)+2\) has a spanning trail or \(G\) is the graph obtained from \(K_{1,3}\) by replacing at most two vertices of degree 1 in \(K_ {1,3}\) with a complete graph or \(G\) is the graph obtained from \(K_{3}\) by adding a pendent edge to each vertex of \(K_{3}\). As a byproduct, we obtain that the line graph of a connected graph \(G\) with \(\alpha (G)\le \kappa (G)+2\) is traceable. These results are all best possible.