Nordhaus–Gaddum type inequality for the integer k-matching number of a graph

Abstract

An integer k-matching of a graph G is a function h:E(G)→{0,1,…,k} such that ∑e∈Γ(v)h(e)≤k for any v∈V(G), where Γ(v) is the set of edges incident to v. The integer k-matching number of G, denoted by mk(G), is the maximum number of ∑e∈E(G)h(e) over all integer k-matching h of G. In this paper, we establish the following lower bounds on the sum of the integer k-matching number of a graph G and its complement by using Gallai–Edmonds Structure Theorem: (1) mk(G)+mk(G¯)≥⌊nk2⌋ for n≥2;(2) if G and G¯ are non-empty, then for n≥25, mk(G)+mk(G¯)≥⌊nk+k2⌋;(3) if G and G¯ have no isolated vertices, then for n≥25, mk(G)+mk(G¯)≥⌊nk2⌋+2k . Furthermore, all extremal graphs attaining the lower bounds are also characterized.