Supereulerian graphs and Chvatal-Erdos type conditions

In 1972, Chvatal and Erdos showed that the graph G with independence number alpha(G) no more than its connectivity k(G) (i.e. k(G) >= alpha(G)) is hamiltonian. In this paper, we consider a kind of Chvatal and Erdos type condition on edge-connectivity (lambda(G)) and matching number (edge independence number). We show that if lambda(G) >= alpha'(G) - 1, then G is either supereulerian or in a well-defined family of graphs. Moreover, we weaken the condition k(G) >= alpha(G) - 1 in [11] to lambda(G) >= alpha(G) - 1 and obtain the similar characterization on non-supereulerian graphs. We also characterize the graph which contains a dominating closed trail under the assumption lambda(G) >= alpha'(G) - 2.