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

Abstract

Tuza’s Conjecture asserts that the minimum number $$\tau _{\varDelta }'(G)$$ of edges of a graph G whose deletion results in a triangle-free graph is at most 2 times the maximum number $$\nu _{\varDelta }'(G)$$ 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 $$\tau _{\varDelta }'(G)$$ and $$\nu _{\varDelta }'(G)$$ into an equivalent one for the clique cover number and the independence number of the triangle graph of G and we provide $$\theta $$ -bounding functions for classes related to triangle graphs. In particular, we obtain optimal $$\theta $$ -bounding functions for the classes $${\textit{Free}}(K_{5}, \text{ claw, } \text{ diamond })$$ and $${\textit{Free}}(P_{5}, \text{ diamond }, K_{2,3})$$ and a $$\chi $$ -bounding function for the class $$(\text{ banner, } \text{ odd-hole }, \overline{K_{1, 4}})$$ .