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

Edge Triangle Packing and Edge Triangle Covering are dual problems extensively studied in the field of parameterized complexity. Given a graph G and an integer k, Edge Triangle Packing seeks to determine whether there exists a set of at least k edge-disjoint triangles in G, while 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 Edge Triangle Packing has a kernel of \((3+\epsilon )k\) vertices, while 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.