Cubic Factor-Invariant Graphs of Bialternating Cycle Quotient Type

Abstract In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph Γ and a vertex-transitive subgroup G of Aut () Aut (Γ), a 2-factor C C of Γ is said to be G -invariant if the set C C is preserved by each element of G. Investigations of factor-invariant cubic graphs therefore contribute to the rapidly growing theory on cubic vertex-transitive graphs, providing a better insight into the structure of such graphs. Initially, the examples where C C consists of a single or just two cycles were analyzed. In a recent paper by Alspach and the author of this paper, the investigation of the examples for which the corresponding quotient graph _ C Γ C of Γ with respect to C C is a cycle was initiated. Moreover, the graphs of the so-called alternating cycle quotient type were classified. In this paper, the remaining examples, that is the graphs of the bialternating cycle quotient type, are classified. It is shown that they belong to a previously unknown infinite 5-parametric family of graphs of girth at most 10 and that they are Cayley graphs of groups with respect to three involutions.