Spanning Eulerian Subgraphs of Large Size

Abstract

A graph $G=(V(G), E(G))$ is supereulerian if it has a spanning Eulerian subgraph. Let $\ell (G)$ be the maximum number of edges of spanning Eulerian subgraphs of a graph G. Motivated by a conjecture due to Catlin on supereulerian graphs, it was shown that if G is an r-regular supereulerian graph, then $\ell (G)\ge \frac{2}{3}|E(G)|$ when $r\ne 5$, and $\ell (G)> \frac{3}{5}|E(G)|$ when $r=5$. In this paper we improve the coefficient and prove that if G is a 5-regular supereulerian graph, then $\ell (G)\ge \frac{19}{30}|E(G)|+\frac{4}{3}$. For this, we first show that each graph G with maximum degree at most 3 has a matching with at least $\frac{2}{7}|E(G)|$ edges and this bound is sharp. Moreover, we show that Catlin’s conjecture holds for claw-free graphs having no vertex of degree 4. Indeed, Catlin’s conjecture does not hold for claw-free graphs in general.