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arXiv · 2411.09806

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

Jan Goedgebeur·Davide Mattiolo·Giuseppe Mazzuoccolo·Jarne Renders·Luca Toffanetti·Isaak H. Wolf
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Abstract

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 for the case where even cycles are included.

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Jan Goedgebeur, Davide Mattiolo, Giuseppe Mazzuoccolo, Jarne Renders, Luca Toffanetti, Isaak H. Wolf. 2024-11-14. On the existence of factors intersecting sets of cycles in regular graphs. https://arxiv.org/abs/2411.09806

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