Comparing the $p$-independence number of regular graphs to the $q$-independence number of their line graphs

TL;DR

This work broadens the classical relation between independence and matching by studying when $\alpha_p(G) \leq \alpha_q(L(G))$ for all $r$-regular graphs. It develops a general framework of valid $\alpha$-triples and proves large families of such triples, notably for $q\ge3$ and for all $r\ge2$, as well as many cases with $q=2$ under parity and size constraints. The results hinge on structural decompositions like $2$-factors and reduced $[k-1,k]$-factors, connecting generalized independence in $G$ to independence in its line graph via $q$-matching arguments. The paper also links unresolved cases to major conjectures in graph theory, namely Linear Arboricity and Path-Cover, suggesting that resolving those conjectures would settle many remaining triples. Overall, it advances understanding of how line graph transformations interact with generalized independence in regular graphs and outlines several directions for future investigation.

Abstract

Let $G$ be a simple graph and let $L(G)$ denote the \emph{line graph} of $G$. A \emph{$p$-independent} set in $G$ is a set of vertices $S \subseteq V(G)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The \emph{$p$-independence number} of $G$, denoted by $α_p(G)$, is the cardinality of a maximum $p$-independent set in $G$. In this paper, and motivated by the recent result that independence number is at most matching number for regular graphs~\cite{CaDaPe2020}, we investigate which values of the non-negative integers $p$, $q$, and $r$ have the property that $α_p(G) \leq α_q(L(G))$ for all r-regular graphs. Triples $(p, q, r)$ having this property are called \emph{valid $α$-triples}. Among the results we prove are: \begin{itemize} \item $(p, q, r)$ is valid $α$-triple for $p \geq 0$, $q \geq 3$ , and $r\geq 2$. \item $(p, q, r)$ is valid $α$-triple for $p \leq q < 3$ and $r\geq 2$. \item $(p, q, r)$ is valid $α$-triple for $p \geq 0$, $q = 2$, and $r$ even. \item $(p, q, r)$ is valid $α$-triple for $p \geq 0$, $q = 2$, and $r$ odd with $r = \max \Big \{ 3, \frac{17(p+1)}{16}\Big \}$. \end{itemize} We also show a close relation between undetermined possible valid $α$-triples, the Linear Aboricity Conjecture, and the Path-Cover Conjecture.

Theorems & Definitions (30)

• Theorem A: CaDaPe2020
• Corollary A: CaDaPe2020
• Theorem B
• Theorem 1: Petersen
• Theorem 2: Tutte
• Lemma 3
• Lemma 4
• Theorem 5: Kano
• Lemma 6
• Theorem 7: tight-matching-bound