Classification of 2-extendable bipartite and cubic non-bipartite vertex-transitive graphs

TL;DR

This paper characterizes 2-extendable bipartite and cubic non-bipartite vertex-transitive graphs, providing precise conditions based on girth and graph isomorphisms, thus advancing the understanding of graph extendability.

Contribution

It offers a complete classification of 2-extendable bipartite Cayley graphs and cubic non-bipartite Cayley graphs, including specific girth and isomorphism conditions.

Findings

Bipartite Cayley graphs are 2-extendable iff not a cycle.

Cubic non-bipartite Cayley graphs with girth ≥ 4 are 2-extendable unless isomorphic to specific graphs.

The Petersen graph is explicitly excluded from 2-extendability in the classification.

Abstract

In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is 2-extendable if and only if it is not a cycle. It is known that a non-bipartite Cayley (vertex-transitive) graph is 2-extendable when it is of minimum degree at least five \cite{sun}. We next classify all 2-extendable cubic non-bipartite Cayley graphs and obtain that: a cubic non-bipartite Cayley graph with girth $g$ is 2-extendable if and only if $g\geq 4$ and it doesn't isomorphic to $Z_{4n}(1,4n-1,2n)$ or $Z_{4n+2}(2,4n,2n+1)$ with $n\geq 2$. Indeed, we prove a more stronger result that a cubic non-bipartite vertex-transitive graph with girth $g$ is 2-extendable if and only if $g\geq 4$ and it doesn't isomorphic to $Z_{4n}(1,4n-1,2n)$ or $Z_{4n+2}(2,4n,2n+1)$ with $n\geq 2$ or the Petersen graph.