Sufficient Conditions for Tuza’s Conjecture on Packing and Covering Triangles

Abstract

Given a simple graph $G=(V,E)$, a subset of E is called a triangle cover if it intersects each triangle of G. Let $\nu _t(G)$ and $\tau _t(G)$ denote the maximum number of pairwise edge-disjoint triangles in G and the minimum cardinality of a triangle cover of G, respectively. Tuza conjectured in 1981 that $\tau _t(G)/\nu _t(G)\le 2$ holds for every graph G. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza’s conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr {T}_G$ of triangles covers all edges in G. We show that a triangle cover of G with cardinality at most $2\nu _t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nu _t(G)/|\mathscr {T}_G|\ge \frac{1}{3}$, (ii) $\nu _t(G)/|E|\ge \frac{1}{4}$, (iii) $|E|/|\mathscr {T}_G|\ge 2$.

Research supported in part by NNSF of China under Grant No. 11531014 and 11222109.