Closing paths to cycles in symmetric graphs

Abstract

It was shown by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in edge-transitive graphs, where the cycle can be assumed to be induced if the path is induced. Motivated by these results, we consider the following problem: For a given class of graphs, determine all integers $\ell\geq 0$ such that for every graph in the class, every path of length at most $\ell$ closes to a cycle. We also consider the variant of the problem for induced paths closing to induced cycles. We completely solve these problems for the classes of (finite) vertex-transitive graphs, edge-transitive graphs, and edge-transitive graphs that are not stars. For all but one case of a negative answer, we provide infinite families of connected counterexamples.