Triangle Transversals in Line Graphs: An Almost-Exact Degree Bound and Tuza's Inequality

For a finite simple graph H, let τ△(L(H)) and ν△(L(H)) denote the minimum number of edges meeting every triangle of its line graph and the maximum number of pairwise edge-disjoint triangles, respectively. Define c(d) = d(d−1)/2 − ⌊d²/4⌋ and B(H) = Σ c(deg_H(v)), where the sum ranges over all vertices of H. We prove the almost-exact bound B(H) ≤ τ△(L(H)) ≤ B(H) + q(H), where q(H) counts the connected components of H that contain a triangle, have only even vertex degrees, and have an odd number of edges. The additive term is sharp. An explicit construction also establishes τ△(L(H)) ≤ 2ν△(L(H)) for every finite simple H. Equality holds precisely when H is triangle-free and all vertex degrees belong to {0, 1, 2, 4, 5}. The proof is elementary, self-contained, and constructive, yielding packing–transversal certificates without optimization.