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

Given a simple graph G = (V,E), a subset of E is called a triangle cover if it intersects each triangle of G. Let νt(G) and τt(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 τt(G)/νt(G) ≤ 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 TG of triangles covers all edges in G. We show that a triangle cover of G with cardinality at most 2νt(G) can be found in polynomial time if one of the following conditions is satisfied: (i) νt(G)/|TG| ≥ 1 3 , (ii) νt(G)/|E| ≥ 1 4 , (iii) |E|/|TG| ≥ 2.