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​∣≥31​, (ii) νt​(G)/∣E∣≥41​, (iii)
∣E∣/∣TG​∣≥2.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm