  In this paper, we study a primal and dual relationship about triangles: For
any graph $G$, let $\nu(G)$ be the maximum number of edge-disjoint triangles in
$G$, and $\tau(G)$ be the minimum subset $F$ of edges such that $G \setminus F$
is triangle-free. It is easy to see that $\nu(G) \leq \tau(G) \leq 3 \nu(G)$,
and in fact, this rather obvious inequality holds for a much more general
primal-dual relation between $k$-hyper matching and covering in hypergraphs.
Tuza conjectured in $1981$ that $\tau(G) \leq 2 \nu(G)$, and this question has
received attention from various groups of researchers in discrete mathematics,
settling various special cases such as planar graphs and generalized to bounded
maximum average degree graphs, some cases of minor-free graphs, and very dense
graphs. Despite these efforts, the conjecture in general graphs has remained
wide open for almost four decades.
  In this paper, we provide a proof of a non-trivial consequence of the
conjecture; that is, for every $k \geq 2$, there exist a (multi)-set $F
\subseteq E(G): |F| \leq 2k \nu(G)$ such that each triangle in $G$ overlaps at
least $k$ elements in $F$. Our result can be seen as a strengthened statement
of Krivelevich's result on the fractional version of Tuza's conjecture (and we
give some examples illustrating this.) The main technical ingredient of our
result is a charging argument, that locally identifies edges in $F$ based on a
local view of the packing solution. This idea might be useful in further
studying the primal-dual relations in general and the Tuza's conjecture in
particular.
