Characterization of forbidden subgraphs for the existence of even factors in a graph

In 2008, it is shown that if H≅K1,3 (the claw), then every H-free graph G has an even factor if and only if δ(G)≥2 and every odd branch-bond of G having an edge-branch. In this paper, we characterize all connected graphs H with order at least three such that every H-free graph has an even factor if and only if δ(G)≥2 and every odd branch-bond of G has an edge-branch.