The minimum number of triangles covering the edges of a graph

Suppose that a connected graph G has n vertices and m edges, and each edge is contained in some triangle of G. Bounds are established here on the minimum number t_min(G) of triangles that cover the edges of G. We prove that ?(n - 1)/2? ? t_min(G) with equality attained only by 3-cactii and by strongly related graphs. We obtain sharp upper bounds: if G is not a 4-clique, then. The triangle cover number t_min(G) is also bounded above by Γ(G) = m - n + c, the cyclomatic number of a graph G of order n with m edges and c connected components. Here we give a combinatorial proof for t_min(G) ? Γ(G) and characterize the family of all extremal graphs satisfying equality.