Characterizations of graphs G having all [1, k]-factors in kG.

Let k≥1 be an integer and G be a graph. Let kG denote the graph obtained from G by replacing each edge of G with k parallel edges. We say that G has all [1,k]-factors or all fractional [1,k]-factors if G has an h-factor or a fractional h-factor for every function h:V(G)→{1,2,…,k} with h(V(G)) even. In this note, we come up with simple characterizations of a graph G such that kG has all [1,k]-factors or all fractional [1,k]-factors. These characterizations are extensions of Tutte’s 1-Factor Theorem and Tutte’s Fractional 1-Factor Theorem.