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

A recent result by Kardoš, Máčajová and Zerafa [J. Comb. Theory, Ser. B. 160 (2023) 1--14] related to the famous Berge-Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say \(\mathcal O\), in a bridgeless cubic graph, there exists a \(1\)-factor intersecting all cycles in \(\mathcal O\) 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 \(\mathcal O\). First of all, we show that \(G\) needs to be \(2\)-connected. Under this additional assumption, we highlight how the ratio \(\frac{t}{r}\) seems to play a crucial role in assuring the existence of a \(t\)-factor \(F\) with the required properties by proving that \(\frac{t}{r} \geq \frac{1}{3}\) is a further necessary condition. We suspect that this condition is also sufficient, and we confirm it in the case \(\frac{t}{r}=\frac{1}{3}\), generalizing the case \(t=1\) and \(r=3\) proved by Kardoš, Máčajová, Zerafa, and in the case \(\frac{t}{r}=\frac{1}{2}\) with \(t\) even. Finally, we provide further results in the case of cycles of arbitrary length.