Triangle Packings and Transversals of Some K4 -Free Graphs

Tuza’s Conjecture asserts that the minimum number τΔ′(G) of edges of a graph G whose deletion results in a triangle-free graph is at most 2 times the maximum number νΔ′(G) of edge-disjoint triangles of G. The complete graphs K4 and K5 show that the constant 2 would be best possible. Moreover, if true, the conjecture would be essentially tight even for K4-free graphs. In this paper, we consider several subclasses of K4-free graphs. We show that the constant 2 can be improved for them and we try to provide the optimal one. The classes we consider are of two kinds: graphs with edges in few triangles and graphs obtained by forbidding certain odd-wheels. We translate an approximate min-max relation for τΔ′(G) and νΔ′(G) into an equivalent one for the clique cover number and the independence number of the triangle graph of G and we provide θ-bounding functions for classes related to triangle graphs. In particular, we obtain optimal θ-bounding functions for the classes Free(K5,claw,diamond) and Free(P5,diamond,K2,3) and a χ-bounding function for the class (banner,odd-hole,K1,4¯).