Packing and covering triangles in bilaterally-complete tripartite graphs (Q6640953)

Let $T$ denote the set of all triangles in $G=(V,E)$. A subset $E^\prime\subseteq E$ is said to be a $T$-transversal if every triangle in $T$ has an edge in $E^\prime$. A subset $T^\prime\subseteq T$ is said to be a packing in $G$ if all the triangles in $T^\prime$ are pairwise edge-disjoint. The cardinality of the smallest $T$-transversal in $G$ is denoted by $\tau_1(G)$ and the cardinality of the largest packing by $\nu_2(G)$. \textit{Z. Tuza} [``Conjecture'', in: A. Hajnal (ed.) et al., Finite and infinite sets. Proceedings of the 6th Hungarian combinatorial colloquium, Eger, 1981. Amsterdam etc.: North-Holland 1984. 888 (1984)] conjectured that $\tau_1(G)\leq 2\nu_2(G)$ for every graph $G$. \textit{S. Aparna Lakshmanan} et al. [Graphs Comb. 28, No. 3, 381--392 (2012; Zbl 1256.05182)] proved that equality holds for certain classes of tripartite graphs. The authors show in this article that this equality also holds for tripartite graphs with only two complete bipartite sides with the help of Menger's theorem and König's line colouring theorem.