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.