On the minimum vertex cover of snarks

Abstract

A vertex cover of a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that every edge of $G$ is incident to at least one vertex in $C$. The vertex cover number of $G$ is the minimum cardinality of a vertex cover of $G$ and is denoted by $τ(G)$. A snark is a connected, bridgeless, cubic graph that has an edge chromatic number of four, meaning its edges cannot be properly colored with only three colors. In this work, we investigate the problem of determining the value of a minimum vertex cover for classes of snark graphs. Given a positive integer $k$, we firstly prove that determining whether an arbitrary snark has a vertex cover $C$ with size $|C| \leq k$ is an NP-complete problem. Secondly, we determine the vertex cover number $τ(G)$ for several subclasses of snark graphs, such as Flower snarks, Goldberg snarks, Generalized Blanuša snarks and Loupekine snarks.