Counterexamples to a conjecture on matching Kneser graphs

Abstract

Let $G$ be a graph and $r\in\mathbb{N}$. The matching Kneser graph $\textsf{KG}(G, rK_2)$ 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 [Alishahi, M. and Hajiabolhassan, H., On the Chromatic Number of Matching Kneser Graphs, Combin. Probab. and Comput. 29 (2020), no. 1, 1--21.] it was conjectured that for any connected graph $G$ and positive integer $r\geq 2$, the chromatic number of $\textsf{KG}(G, rK_2)$ is equal to $|E(G)|-\textsf{ex}(G,rK_2)$, where $\textsf{ex}(G,rK_2)$ 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.