Spanning eulerian subgraphs and matchings

AbstractLet G be a 2-edge-connected graph of order n. For a matching M3 consisting of three independent edges of E(G), let Σ(M3) denote the sum of the degrees of the six vertices incident with M3. We show that if Σ(M3)⩾2n+2 for all 3-matching M3 of G, then either G has a spanning eulerian subgraph, or there is a connected subgraph H of G such that the contraction G/H is K2,t for some odd t. We describe the nature of this contraction. The inequality is best-possible. We obtain several previous results as special cases