A note on minimum degree conditions for supereulerian graphs Original Research Article

A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut S⊆E(G) with |S|⩽3 satisfies the property that each component of G−S has order at least (n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ⩾4: If G is a 2-edge-connected graph of order n with δ(G)⩾4 such that for every edge xy∈E(G) , we have max{d(x),d(y)}⩾(n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ(G)⩾4 cannot be relaxed.