On extremal -supereulerian graphs

A graph G is called k -supereulerian if it has a spanning even subgraph with at most k components. In this paper, we prove that any 2-edge-connected loopless graph of order n is ⌈ ( n − 2 ) / 3 ⌉ -supereulerian, with only one exception. This result solves a conjecture in [Z. Niu, L. Xiong, Even factor of a graph with a bounded number of components, Australas. J. Combin. 48 (2010) 269–279]. As applications, we give a best possible size lower bound for a 2-edge-connected simple graph G with n > 5 k + 2 vertices to be k -supereulerian, a best possible minimum degree lower bound for a 2-edge-connected simple graph G such that its line graph L ( G ) has a 2-factor with at most k components, for any given integer k > 0 , and a sufficient condition for k -supereulerian graphs.