Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell [The spanning subgraphs of Eulerian graphs, J. Graph Theory 1 (1977), pp. 79-84] proposed the supereulerian graph problem which seeks the characterization of graphs with a spanning Eulerian subgraph. Pulleyblank [A note on graphs spanned by Eulerian graphs, J. Graph Theory 3 (1979), pp. 309-310] showed that the supereulerian problem, even within planar graphs, is NP-complete. In this paper, we settle an open problem raised by An and Xiong on characterization of supereulerian graphs with small matching numbers. A wellknown theorem by Chvatal and Erdos [A note on Hamilton circuits, Discrete Math. 2 (1972), pp. 111-135] states that if G satisfies alpha(G) <= kappa G), then G is hamiltonian. Flandrin and Li in 1989 showed that every 3-connected claw-free graph G with alpha(G) <= 2 kappa(G) is hamiltonian. Our characterization is also applied to show that every 2-connected claw-free graph G with alpha(G) <= 3 is hamiltonian, with only one well-characterized exceptional class.