Large Even Factors Of A Graph

For a graph G,e(G)denotes the number of edges in G.Assume that G is a simple graph with exactly 2k vertices of odd degrees for an integer k? 1.An even factor of G is a spanning even subgraph in which each vertex has positive even degree.We conjecture that if G has an even factor,then it has an even factor F with e(F)?2/3(e(G)-k+ 2)).We show that it is true for k ? {1,2}.Moreover,for the case k = 1,if e(H)?2/3(e(G)+ 1)),for every even factor H of G,then G belongs to a specified class of graphs.