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

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 König--Egerváry graphs, a generalization of the well-studied König--Egerváry graphs.

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