A note on minimum degree conditions for supereulerian graphs

Abstract

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.