Triangle Packings and Transversals of Some $$K_{4}$$ K 4 -Free Graphs

Abstract

Tuza's Conjecture asserts that the minimum number τ ∆ (G) of edges of a graph G whose deletion results in a triangle-free graph is at most 2 times the maximum number ν ∆ (G) of edge-disjoint triangles of G. The complete graphs K 4 and K 5 show that the constant 2 would be best possible. Moreover, if true, the conjecture would be essentially tight even for K 4-free graphs. In this paper, we consider several subclasses of K 4-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 τ ∆ (G) and ν ∆ (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(K 5 , claw, diamond) and Free(P 5 , diamond, K 2,3) and a χ-bounding function for the class (banner, odd-hole, K 1,4).