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

Abstract

Abstract Let k ≥ 1 be an integer and G be a graph. Let k G 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 k G 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.