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

Abstract

Let $G$ be a simple graph and let $L\left(G\right)$ denote the line graph of $G$. A $p$-independent set in $G$ is a set of vertices $S\subseteq V\left(G\right)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The $p$-independence number of $G$, denoted by ${\alpha }_p\left(G\right)$, 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 Caro et al., (2022), we investigate which values of the non-negative integers $p$, $q$, and $r$ have the property that ${\alpha }_p\left(G\right)\le {\alpha }_q\left(L\left(G\right)\right)$ for all r-regular graphs. Triples $(p,q,r)$ having this property are called valid $\alpha$-triples. Among the results we prove are:

• •
  $(p,q,r)$ is valid $\alpha$-triple for $p\ge 0$, $q\ge 3$ , and $r\ge 2$.
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  $(p,q,r)$ is valid $\alpha$-triple for $p\le q<3$ and $r\ge 2$.
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  $(p,q,r)$ is valid $\alpha$-triple for $p\ge 0$, $q=2$, and $r$ even.
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  $(p,q,r)$ is valid $\alpha$-triple for $p\ge 0$, $q=2$, and $r$ odd with $r=max\left\{3,\frac{17(p+1)}{16}\right\}$.

We also show a close relation between undetermined possible valid $\alpha$-triples, the Linear Aboricity Conjecture, and the Path-Cover Conjecture.