Fan-Type Conditions for Spanning Eulerian Subgraphs

For a graph G, let dF (G) = min{max{d(u), d(v)}| for any u, v. V(G) with distance 2}. A graph is supereulerian if it has a spanning Eulerian subgraph. Let p > 0, g > 2 and be given nonnegative numbers. Let Q be the family of nonsupereulerian 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 dF (G) = n (g-2) p - and n is sufficiently large, then G is supereulerian if and only if G is not contractible to a graph inQ. Results on best possible values of p and for such graphs to either be supereulerian or be contractible to the Petersen graph are given.