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\textsc{Edge Triangle Packing} and \textsc{Edge Triangle Covering} are dualproblems extensively studied in the field of parameterized complexity.  Given a graph $G$ and an integer $k$, \textsc{Edge Triangle Packing} seeks todetermine whether there exists a set of at least $k$ edge-disjoint triangles in$G$,  while \textsc{Edge Triangle Covering} aims to find out whether there exists aset of at most $k$ edges that intersects all triangles in $G$.  Previous research has shown that \textsc{Edge Triangle Packing} has a kernelof $(3+\epsilon)k$ vertices, while \textsc{Edge Triangle Covering} has a kernelof $6k$ vertices.  In this paper, we show that the two problems allow kernels of $3k$ vertices,improving all previous results. A significant contribution of our work is theutilization of a novel discharging method for analyzing kernel size, whichexhibits potential for analyzing other kernel algorithms.

A Discharging Method: Improved Kernels for Edge Triangle Packing and Covering