\textsc{Edge Triangle Packing} and \textsc{Edge Triangle Covering} are dual problems extensively studied in the field of parameterized complexity. Given a graph $G$ and an integer $k$, \textsc{Edge Triangle Packing} seeks to determine 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 a set of at most $k$ edges that intersects all triangles in $G$. Previous research has shown that \textsc{Edge Triangle Packing} has a kernel of $(3+\epsilon)k$ vertices, while \textsc{Edge Triangle Covering} has a kernel of $6k$ 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 the utilization of a novel discharging method for analyzing kernel size, which exhibits potential for analyzing other kernel algorithms.

翻译：\textsc{边三角形打包}（Edge Triangle Packing）和\textsc{边三角形覆盖}（Edge Triangle Covering）是参数化复杂度领域中广泛研究的对偶问题。给定图$G$和整数$k$，\textsc{边三角形打包}旨在判断$G$中是否存在至少$k$个边不相交的三角形，而\textsc{边三角形覆盖}则试图找出是否存在至多$k$条边的集合，该集合与$G$中所有三角形相交。先前研究表明，\textsc{边三角形打包}具有$(3+\epsilon)k$个顶点的核，而\textsc{边三角形覆盖}具有$6k$个顶点的核。本文证明这两个问题均允许$3k$个顶点的核，改进了所有先前结果。我们工作的一个重要贡献是采用了一种新颖的放电方法来分析核大小，该方法在分析其他核算法方面展现出潜力。