Fan-Type Conditions for Spanning Eulerian Subgraphs

For a graph $$G$$G, let $$\delta _F(G)=\min \{\max \{d(u), d(v)\} | \text{ for } \text{ any }~u, v\in V(G)\, \text{ with } \text{ distance }~2\}$$δF(G)=min{max{d(u),d(v)}|foranyu,v∈V(G)withdistance2}. A graph is supereulerian if it has a spanning Eulerian subgraph. Let $$p>0$$p>0, $$g>2$$g>2 and $$\epsilon $$ϵ be given nonnegative numbers. Let $$\mathcal{Q}$$Q be the family of non-supereulerian graphs with order at most $$5(p-2)$$5(p-2). In this paper, we prove that for a 3-edge-connected graph $$G$$G of order $$n$$n, if $$G$$G satisfies a Fan-type condition $$\delta _F(G)\ge \frac{n}{(g-2)p}-\epsilon $$δF(G)≥n(g-2)p-ϵ and $$n$$n is sufficiently large, then $$G$$G is supereulerian if and only if $$G$$G is not contractible to a graph in $$\mathcal{Q}$$Q. Results on best possible values of $$p$$p and $$\epsilon $$ϵ for such graphs to either be supereulerian or be contractible to the Petersen graph are given.