On covering cubic graphs with three perfect matchings

TL;DR

The paper investigates the range of values that the invariant $m_3(G)$ can take for bridgeless cubic graphs, where $m_3(G)$ is the maximal fraction of edges coverable by three perfect matchings. It develops multipole-based constructions to realize exact values of $m_3$; first for cyclic connectivity $2$ via graphs $G_{a,b}$ with $m_3(G_{a,b})=(4a+2b)/(5a+2b)$, and then for cyclic connectivity $4$ via $G^4_{a,b}$ with $m_3(G^4_{a,b})=(9a+4b)/(10a+4b)$. By suitable choice of parameters, the authors show there exist infinitely many $2$-connected cubic graphs with any $m_3=p/q\in[4/5,1)$ and infinitely many cyclically $4$-connected cubic graphs with any $m_3=p/q\in[9/10,1)$, providing explicit constructions and bounds. They also identify a $28$-vertex snark with $m_3=37/42$, establishing a concrete lower bound $c_4$ for the $k=4$ case, and give a detailed framework using Blanuša blocks and I-extensions to justify the connectivity and exact cover properties of the new gadgets.

Abstract

For a bridgeless cubic graph $G$, $m_3(G)$ is the ratio of the maximum number of edges of $G$ covered by the union of $3$ perfect matchings to $|E(G)|$. We prove that for any $r\in [4/5, 1)$, there exist infinitely many cubic graphs $G$ such that $m_3(G) = r$. For any $r\in [9/10, 1)$, there exist infinitely many cyclically $4$-connected cubic graphs $G$ with $m_3(G) = r$.

Figures (1)

• Figure 1: The multipole $A'$ used to construct cyclically $4$-connected graphs.

Theorems & Definitions (18)

• Conjecture 1: kaiser
• Theorem 1
• Theorem 2
• Conjecture 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof