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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

Abstract

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.

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

Abstract

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 , in a bridgeless cubic graph, there exists a -factor intersecting all cycles in in at least one edge. This remarkable result opens up natural generalizations in the case of an -regular graph and a -factor , with and being positive integers. In this paper, we start the study of this problem by proving necessary and sufficient conditions on , and to assure the existence of a suitable for any possible choice of the set . First of all, we show that needs to be -connected. Under this additional assumption, we highlight how the ratio seems to play a crucial role in assuring the existence of a -factor with the required properties by proving that is a further necessary condition. We suspect that this condition is also sufficient, and we confirm it in the case , generalizing the case and proved by Kardoš, Máčajová, Zerafa, and in the case with even. Finally, we provide further results for the case where even cycles are included.

Paper Structure

This paper contains 7 sections, 18 theorems, 8 figures.

Table of Contents

  1. Introduction
  2. First necessary condition: 2-connectivity
  3. Second necessary condition: t/r >= 1/3
  4. A first sufficient condition: t/r=1/3
  5. Another sufficient condition: t/r=1/2, t even.
  6. Arbitrary cycles
  7. Concluding remarks and open problems

Key Result

Theorem 1.5

Let $G$ be a 2-connected cubic graph. Let $\mathcal{O}$ be a set of pairwise edge-disjoint odd cycles of $G$ and let $e \in E(G)$. Then, there exists a 1-factor $F$ of $G$ such that $e \in E(F)$ and $E(F) \cap E(O) \neq \emptyset$ for every $O \in \mathcal{O}$.

Figures (8)

  • Figure 1: The graph $J$ introduced in the proof of Theorem \ref{['theo:1_connected_examples']} (in the case of even $r$ and $t$).
  • Figure 2: The graph $G_6$ introduced in the proof of the Theorem \ref{['theo:1_connected_examples']} (in the case $r=6$ and $t$ even).
  • Figure 3: The paw graph introduced in the proof of the Theorem \ref{['theo:1_connected_examples']} (in the case of even $r$ and odd $t$).
  • Figure 4: The graph $W$ introduced in the proof of the Theorem \ref{['theo:1_connected_examples']} (in the case $t=1$).
  • Figure 5: The graph $G$ constructed in the proof of Theorem \ref{['theo:odd_circuits_t<r/3']} in the case $r=4$. The set $\mathcal{O}$ consists of the triangles whose edges are drawn with dashed lines.
  • ...and 3 more figures

Theorems & Definitions (31)

  • Conjecture 1.1: Fulkerson fulkerson1971blocking
  • Conjecture 1.2: Fan, Raspaud fan1994fulkerson
  • Conjecture 1.3: Máčajová, Škoviera MACAJOVA2005112, see also Kaiser2010
  • Conjecture 1.4: Mazzuoccolo MAZZUOCCOLO2013235
  • Theorem 1.5: Kardoš, Máčajová, Zerafa KARDOS20231
  • Theorem 1.6
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • ...and 21 more