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 for the case where even cycles are included.

翻译：Kardoš、Máčajová 和 Zerafa [J. Comb. Theory, Ser. B. 160 (2023) 1--14] 最近的一项与著名 Berge-Fulkerson 猜想相关的结果表明：在无桥三次图中，给定任意一组两两边不交的奇圈集合 $\mathcal O$，总存在一个 $1$-因子与 $\mathcal O$ 中所有圈至少有一条公共边。这一显著结果为 $r$-正则图 $G$ 与 $t$-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 $G$、$t$ 和 $r$ 满足的充分必要条件来启动该问题的研究，以确保对于任意可能的集合 $\mathcal O$ 均存在合适的 $F$。我们首先证明 $G$ 必须是 $2$-连通的。在此附加假设下，我们指出比值 $\frac{t}{r}$ 对保证具有所需性质的 $t$-因子 $F$ 的存在性似乎起着关键作用，并证明 $\frac{t}{r} \geq \frac{1}{3}$ 是另一个必要条件。我们推测该条件也是充分的，并在 $\frac{t}{r}=\frac{1}{3}$ 的情形下予以证实——这推广了 Kardoš、Máčajová 和 Zerafa 所证明的 $t=1$ 且 $r=3$ 的情形，并在 $\frac{t}{r}=\frac{1}{2}$ 且 $t$ 为偶数的情形下得到验证。最后，我们针对包含偶圈的情形给出了进一步的结果。