Abstract :
Let G be a graph and r ∈ N. The matching Kneser graph KG(G, rK2) is a graph whose vertex set is the set of r-matchings in G and two vertices are adjacent if their corresponding matchings are edge-disjoint. In [M. Alishahi and H. Hajiabolhassan, On the Chromatic Number of Matching Kneser Graphs, Combin. Probab. and Comput., 29, No. 1 (2020), 1–21] it was conjectured that for any connected graph G and positive integer r ≥ 2, the chromatic number of KG(G, rK2) is equal to |E(G)| − ex(G, rK2), where ex(G, rK2) denotes the largest number of edges in G avoiding a matching of size r. In this note, we show that the conjecture is not true for snarks.
