On the ratio between the maximum weight of a perfect matching and the maximum weight of a matching

Abstract

Let G be a finite graph (without loops and multiple edges) and let w:E(G)→[0,+∞) be a weight function on the edge set of G. We consider the ratio between the maximum weight of a perfect matching of G and the maximum weight of a matching of G. The parameter η(G), introduced by Brazil & al. in Brazil et al. (2016), is defined as the minimum of such a ratio among all nonnegative edge weight assignments of G. In the present paper, we propose a way to compute a lower bound for the parameter η(G), and we use it to prove that for every rational number q in the interval [0,1] there exists a graph G such that η(G)=q. Moreover, we further use the same method, in combination with some new arguments, to establish the value of η for Prism graphs and Möbius Ladders. Finally, we improve known results for Blanuša Snarks B1 and B2 by determining the exact value of η(B1) and η(B2).