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

Suppose that I wish to make a graph G triangle-free by removing a small number of edges. An obvious obstruction is the presence of a large set of edge-disjoint triangles, since I must remove one edge from each triangle. On the other hand, removing all the edges in a maximal set of edge-disjoint triangles clearly makes G triangle-free. Tuza’s Conjecture states that the worstcase number of edges that must be removed is somewhere between these extremes: if τ(G) is the number of edges that must be removed to make G triangle-free and ν(G) is the maximum number of edge-disjoint triangles in G, then Tuza’s Conjecture states that τ(G) ≤ 2ν(G). Using the discharging method, we show that Tuza’s Conjecture holds whenever G has no subgraph with maximum average degree at least 7. This subsumes several earlier results and represents the first application of discharging to this problem.