On the Chromatic Number of Matching Kneser Graphs

Abstract

AbstractIn an earlier paper, the present authors (2015) introduced thealtermatic numberof graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.