Integer [formula omitted]-matchings of graphs: [formula omitted]-Berge–Tutte formula, [formula omitted]-factor-critical graphs and [formula omitted]-barriers

Abstract

An integer k-matching of a graph G is a function h that assigns to each edge an integer in {0,…,k} such that ∑e∈Γ(v)h(e)≤k for each vertex v, where Γ(v) is the set of edges incident with v. The integer k-matching number of G is the maximum number of ∑e∈E(G)h(e) over all integer k-matchings h of G. When k is even, Yan Liu and Xiaohui Liu proved that the integer k-matching number of G equals k times its fractional matching number. In this paper, when k is odd, we prove the integer k-matching analogue of the Berge-Tutte Formula, define k-factor-critical graph, k-barrier and k-extreme, respectively, give two sufficient and necessary conditions of k-factor-critical graph which is similar to those of factor-critical graph, and obtain some properties about k-barrier and k-extreme, respectively.