Favaron's Theorem, k-dependence, and Tuza's Conjecture

TL;DR

This paper extends Favaron's theorem on $k$-dependent sets in graphs, reveals a stronger structural property of these sets, and uncovers a surprising link to Tuza's conjecture on triangle packing and covering.

Contribution

It generalizes Favaron's theorem, introduces a stronger structural property of $k$-dependent sets, and connects these concepts to Tuza's conjecture.

Findings

Extended Favaron's theorem with a stronger structural property

Established a connection between $k$-dependent sets and Tuza's conjecture

Provided insights into packing and covering of triangles in graphs

Abstract

A vertex set $D$ in a graph $G$ is $k$-dependent if $G[D]$ has maximum degree at most $k-1$, and $k$-dominating if every vertex outside $D$ has at least $k$ neighbors in $D$. Favaron proved that if $D$ is a $k$-dependent set maximizing the quantity $k|D| - |E(G[D])|$, then $D$ is $k$-dominating. We extend this result, showing that such sets satisfy a stronger structural property, and we find a surprising connection between Favaron's theorem and a conjecture of Tuza regarding packing and covering of triangles.