Tuza's Conjecture for Graphs of Maximum Average Degree Less Than 7

TL;DR

This paper proves Tuza's Conjecture for graphs with maximum average degree less than 7 by introducing reducible sets and weak K"onig--Egerváry graphs, advancing understanding of triangle coverings.

Contribution

The paper establishes Tuza's Conjecture for a new class of graphs using novel concepts like reducible sets and weak K"onig--Egerváry graphs.

Findings

Tuza's Conjecture holds for graphs with average degree less than 7.

Introduction of reducible sets as a key tool in the proof.

Definition of weak K"onig--Egerváry graphs as a generalization.

Abstract

Tuza's Conjecture states that if a graph $G$ does not contain more than $k$ edge-disjoint triangles, then some set of at most $2k$ edges meets all triangles of $G$. We prove Tuza's Conjecture for all graphs $G$ having no subgraph with average degree at least $7$. As a key tool in the proof, we introduce a notion of reducible sets for Tuza's Conjecture; these are substructures which cannot occur in a minimal counterexample to Tuza's Conjecture. We also introduce weak K\"onig--Egerv\'ary graphs, a generalization of the well-studied K\"onig--Egerv\'ary graphs.