Supereulerian Graphs and the Petersen Graph

A graph G is called even if O(G)=<, and G is called eulerian if G is even and connected. If G has a spanning eulerian subgraph, then G is called supereulerian, and we write G # SL. Tutte [19, 20] and Matthews [17] conjectured that if a 2-edge-connected graph G has no subgraph contractible to the Petersen graph, then G has a 3-colorable double cycle cover (i.e., a collection of three even subgraphs such that each edge of G lies in exactly two of them). We showed before (see [5 or 6]) that any supereulerian graph has a 3-colorable double cycle cover. In this context, our present result, that any 3-edge-connected graph with at most 10 edge cuts of size 3 is either supereulerian or is itself contractible to the Petersen graph, is of interest. Jaeger [12] had previously shown that any 4-edge-connected graph supereulerian. Catlin [5] recently showed that article no. 0009