Max-cut and extendability of matchings in distance-regular graphs

Abstract

A connected graph G of even order v is called t-extendable if it contains a perfect matching, t<v/2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. Since its introduction by Plummer in the 1980s, this combinatorial parameter has been studied for many classes of interesting graphs. In 2005, Brouwer and Haemers proved that every distance-regular graph of even order is 1-extendable and in 2014, Cioabă and Li showed that any connected strongly regular graph of even order is 3-extendable except for a small number of exceptions.In this paper, we extend and generalize these results. We prove that all distance-regular graphs with diameter D≥3 are 2-extendable and we also obtain several better lower bounds for the extendability of distance-regular graphs of valency k≥3 that depend on k, λ and μ, where λ is the number of common neighbors of any two adjacent vertices and μ is the number of common neighbors of any two vertices in distance two. In many situations, we show that the extendability of a distance-regular graph with valency k grows linearly in k. We conjecture that the extendability of a distance-regular graph of even order and valency k is at least ⌈k/2⌉−1 and we prove this fact for bipartite distance-regular graphs.In course of this investigation, we obtain some new bounds for the max-cut and the independence number of distance-regular graphs in terms of their size and odd girth and we prove that our inequalities are incomparable with known eigenvalue bounds for these combinatorial parameters.