A two-phase optimization method for a multiobjective vehicle relocation problem in electric carsharing systems

A spanning subgraph F of a graph G is called an even factor of G if each vertex of F has even degree at least 2 in F. It was conjectured that if a graph G has an even factor, then it has an even factor F with (|E(F)|ge {4over 7}(|E(G)| + 1)+ {2over 7}|V_2(G)|), where (V_2(G)) is the set of vertices of degree 2 in G. We note that the conjecture is false if G is a triangle. In this paper, we confirm the conjecture for all graphs on at least 4 vertices, and moreover, we prove that if (|E(H)|le {4over 7}(|E(G)| + 1)+ {2over 7}|V_2(G)|) for every even factor H of G, then every maximum even factor of G is a 2-factor consisting of even circuits.