A note on minimum degree conditions for supereulerian graphs
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 ??4: If G is a 2-edge-connected graph of order n with ?(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 ?(G)?4 cannot be relaxed.
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