5,326 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
The edge chromatic number of outer-1-planar graphs
A graph is outer-1-planar if it can be drawn in the plane so that all
vertices are on the outer face and each edge is crossed at most once. In this
paper, we completely determine the edge chromatic number of outer 1-planar
graphs
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Total coloring of 1-toroidal graphs of maximum degree at least 11 and no adjacent triangles
A {\em total coloring} of a graph is an assignment of colors to the
vertices and the edges of such that every pair of adjacent/incident
elements receive distinct colors. The {\em total chromatic number} of a graph
, denoted by \chiup''(G), is the minimum number of colors in a total
coloring of . The well-known Total Coloring Conjecture (TCC) says that every
graph with maximum degree admits a total coloring with at most colors. A graph is {\em -toroidal} if it can be drawn in torus such
that every edge crosses at most one other edge. In this paper, we investigate
the total coloring of -toroidal graphs, and prove that the TCC holds for the
-toroidal graphs with maximum degree at least~ and some restrictions on
the triangles. Consequently, if is a -toroidal graph with maximum degree
at least~ and without adjacent triangles, then admits a total
coloring with at most colors.Comment: 10 page
On edge-group choosability of graphs
In this paper, we study the concept of edge-group choosability of graphs. We
say that G is edge k-group choosable if its line graph is k-group choosable. An
edge-group choosability version of Vizing conjecture is given. The evidence of
our claim are graphs with maximum degree less than 4, planar graphs with
maximum degree at least 11, planar graphs without small cycles, outerplanar
graphs and near-outerplanar graphs
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