On the existence of factors intersecting sets of cycles in regular graphs

A recent result by Kardoš et al. (2023) related to the famous Berge–Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say O, in a bridgeless cubic graph, there exists a 1-factor intersecting all cycles in O in at least one edge. This remarkable result opens up natural generalizations in the case of an r-regular graph G and a t-factor F, with r and t being positive integers. In this paper, we start the study of this problem by proving necessary and sufficient conditions on G, t and r to assure the existence of a suitable F for any possible choice of the set O. First of all, we show that G needs to be 2-connected. Under this additional assumption, we highlight how the ratio tr seems to play a crucial role in assuring the existence of a t-factor F with the required properties by proving that tr≥13 is a further necessary condition. We suspect that this condition is also sufficient, and we confirm it in the case tr=13, generalizing the case t=1 and r=3 proved by Kardoš, Máčajová, Zerafa, and in the case tr=12 with t even. Finally, we provide further results for the case where even cycles are included.