Fan-Type Conditions for Spanning Eulerian Subgraphs

Abstract

For a graph $G$, let $\delta _F(G)=\min \{\max \{d(u), d(v)\} | \text{ for } \text{ any }~u, v\in V(G)\, \text{ with } \text{ distance }~2\}$. A graph is supereulerian if it has a spanning Eulerian subgraph. Let $p>0$, $g>2$ and $\epsilon$ be given nonnegative numbers. Let $\mathcal{Q}$ be the family of non-supereulerian graphs with order at most $5(p-2)$. In this paper, we prove that for a 3-edge-connected graph $G$ of order $n$, if $G$ satisfies a Fan-type condition $\delta _F(G)\ge \frac{n}{(g-2)p}-\epsilon$ and $n$ is sufficiently large, then $G$ is supereulerian if and only if $G$ is not contractible to a graph in $\mathcal{Q}$. Results on best possible values of $p$ and $\epsilon$ for such graphs to either be supereulerian or be contractible to the Petersen graph are given.