A recent result by Kardo\v{s}, M\'a\v{c}ajov\'a 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 Kardo\v{s}, M\'a\v{c}ajov\'a, 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.

翻译：Kardo\v{s}、M\'a\v{c}ajov\'a 和 Zerafa 最近的一项结果 [J. Comb. Theory, Ser. B. 160 (2023) 1--14] 与著名的 Berge-Fulkerson 猜想相关，该结果表明：在无桥三次图中，给定任意一组两两边不交的奇圈集合 $\mathcal O$，存在一个 $1$-因子与 $\mathcal O$ 中所有圈至少在一条边上相交。这一显著结果为 $r$-正则图 $G$ 和 $t$-因子 $F$（其中 $r$ 和 $t$ 为正整数）情形下的自然推广开辟了道路。本文通过证明对任意可能的集合 $\mathcal O$ 确保存在合适 $F$ 时 $G$、$t$ 和 $r$ 所需满足的充要条件，开启了该问题的研究。首先，我们证明 $G$ 必须是 $2$-连通的。在此附加假设下，我们通过证明 $\frac{t}{r} \geq \frac{1}{3}$ 是另一个必要条件，强调了比值 $\frac{t}{r}$ 在确保存在具有所需性质的 $t$-因子 $F$ 时似乎起着关键作用。我们推测该条件也是充分的，并在 $\frac{t}{r}=\frac{1}{3}$ 的情形（推广了 Kardo\v{s}、M\'a\v{c}ajov\'a 和 Zerafa 证明的 $t=1$ 且 $r=3$ 的情形）以及 $\frac{t}{r}=\frac{1}{2}$ 且 $t$ 为偶数的情形中证实了这一推测。最后，我们针对任意长度圈的情形提供了进一步的结果。