454 research outputs found
Minimal counterexamples and discharging method
Recently, the author found that there is a common mistake in some papers by
using minimal counterexample and discharging method. We first discuss how the
mistake is generated, and give a method to fix the mistake. As an illustration,
we consider total coloring of planar or toroidal graphs, and show that: if
is a planar or toroidal graph with maximum degree at most , where
, then the total chromatic number is at most .Comment: 8 pages. Preliminary version, comments are welcom
Domination number of graphs with minimum degree five
We prove that for every graph on vertices and with minimum degree
five, the domination number cannot exceed . The proof combines
an algorithmic approach and the discharging method. Using the same technique,
we provide a shorter proof for the known upper bound on the domination
number of graphs of minimum degree four.Comment: 17 page
A Discharging Method: Improved Kernels for Edge Triangle Packing and Covering
\textsc{Edge Triangle Packing} and \textsc{Edge Triangle Covering} are dual
problems extensively studied in the field of parameterized complexity.
Given a graph and an integer , \textsc{Edge Triangle Packing} seeks to
determine whether there exists a set of at least edge-disjoint triangles in
,
while \textsc{Edge Triangle Covering} aims to find out whether there exists a
set of at most edges that intersects all triangles in .
Previous research has shown that \textsc{Edge Triangle Packing} has a kernel
of vertices, while \textsc{Edge Triangle Covering} has a kernel
of vertices.
In this paper, we show that the two problems allow kernels of 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
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