Graph covers and generalized snarks

Abstract

The notion of graph cover, also known as locally bijective homomorphism, is a discretization of covering spaces known from general topology. It is a pair of incidence-preserving vertex- and edge-mappings between two graphs, the edge-component being bijective on the edge-neighborhoods of every vertex and its image. In line with the current trends in topological graph theory and its applications in mathematical physics, graphs are considered in the most relaxed form and as such they may contain multiple edges, loops and semi-edges. Nevertheless, simple graphs (binary structures without multiple edges, loops, or semi-edges) play an important role. The Strong Dichotomy Conjecture of Bok et al. [2022] states that for every fixed graph $H$, deciding if an input graph covers $H$ is either polynomial time solvable for arbitrary input graphs, or NP-complete for simple ones. These authors introduced the following quasi-order on the class of connected graphs: A connected graph $A$ is called {\em stronger than} a connected graph $B$ if every simple graph that covers $A$ also covers $B$. Witnesses of $A$ not being stronger than $B$ are {\em generalized snarks} in the sense that they are simple graphs that cover $A$ but do not cover $B$. Bok et al. conjectured that if $A$ has no semi-edges, then $A$ is stronger than $B$ if and only if $A$ covers $B$. We prove this conjecture for cubic one-vertex graphs, and we also justify it for all cubic graphs $A$ with at most 4 vertices.