Counterexamples to a conjecture on matching Kneser graphs

TL;DR

This paper disproves a conjecture relating the chromatic number of matching Kneser graphs to the maximum edges avoiding certain matchings, specifically providing counterexamples among snarks.

Contribution

It provides the first known counterexamples to the conjecture, showing it does not hold for all connected graphs, especially snarks.

Findings

The conjecture does not hold for all connected graphs.

Counterexamples are found among snarks.

The chromatic number can differ from the conjectured value.

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.