228 research outputs found
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
Common graphs with arbitrary chromatic number
Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a
sufficiently large complete graph contains a monochromatic copy of H. In 1962,
Erdos conjectured that the random 2-edge-coloring minimizes the number of
monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta
to all graphs. In the late 1980s, the conjectures were disproved by Thomason
and Sidorenko, respectively. A classification of graphs whose number of
monochromatic copies is minimized by the random 2-edge-coloring, which are
referred to as common graphs, remains a challenging open problem. If
Sidorenko's Conjecture, one of the most significant open problems in extremal
graph theory, is true, then every 2-chromatic graph is common, and in fact, no
2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While
examples of 3-chromatic common graphs were known for a long time, the existence
of a 4-chromatic common graph was open until 2012, and no common graph with a
larger chromatic number is known.
We construct connected k-chromatic common graphs for every k. This answers a
question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab.
Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov
[London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This
also answers in a stronger form the question raised by Jagger, Stovicek and
Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common
graph with chromatic number at least four.Comment: Updated to include reference to arXiv:2207.0942
GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs
We present a prototype of a software tool for exploration of multiple
combinatorial optimisation problems in large real-world and synthetic complex
networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial
Explorer), provides a unified framework for scalable computation and
presentation of high-quality suboptimal solutions and bounds for a number of
widely studied combinatorial optimisation problems. Efficient representation
and applicability to large-scale graphs and complex networks are particularly
considered in its design. The problems currently supported include maximum
clique, graph colouring, maximum independent set, minimum vertex clique
covering, minimum dominating set, as well as the longest simple cycle problem.
Suboptimal solutions and intervals for optimal objective values are estimated
using scalable heuristics. The tool is designed with extensibility in mind,
with the view of further problems and both new fast and high-performance
heuristics to be added in the future. GraphCombEx has already been successfully
used as a support tool in a number of recent research studies using
combinatorial optimisation to analyse complex networks, indicating its promise
as a research software tool
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