126 research outputs found
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
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
Advances in Discrete Applied Mathematics and Graph Theory
The present reprint contains twelve papers published in the Special Issue âAdvances in Discrete Applied Mathematics and Graph Theory, 2021â of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs
Balanced-chromatic number and Hadwiger-like conjectures
Motivated by different characterizations of planar graphs and the 4-Color
Theorem, several structural results concerning graphs of high chromatic number
have been obtained. Toward strengthening some of these results, we consider the
\emph{balanced chromatic number}, , of a signed graph
. This is the minimum number of parts into which the vertices of a
signed graph can be partitioned so that none of the parts induces a negative
cycle. This extends the notion of the chromatic number of a graph since
, where denotes the signed graph
obtained from~ by replacing each edge with a pair of (parallel) positive and
negative edges. We introduce a signed version of Hadwiger's conjecture as
follows.
Conjecture: If a signed graph has no negative loop and no
-minor, then its balanced chromatic number is at most .
We prove that this conjecture is, in fact, equivalent to Hadwiger's
conjecture and show its relation to the Odd Hadwiger Conjecture.
Motivated by these results, we also consider the relation between
subdivisions and balanced chromatic number. We prove that if has
no negative loop and no -subdivision, then it admits a balanced
-coloring. This qualitatively generalizes a result of
Kawarabayashi (2013) on totally odd subdivisions
Signed graphs, regular matroids, grafts
Mathematics
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