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, 2019
We consider three colouring problems which are variations of the basic vertex-colouring problem, and are motivated by applications from various domains. We give pointers to theoretical and algorithmic developments for each of these variations.
A. Hertz, B. Ries
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We consider three colouring problems which are variations of the basic vertex-colouring problem, and are motivated by applications from various domains. We give pointers to theoretical and algorithmic developments for each of these variations.
A. Hertz, B. Ries
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The improved ColourAnt algorithm: a hybrid algorithm for solving the graph colouring problem
International Journal of Bio-Inspired Computation (IJBIC), 2020The graph colouring problem is interesting because of its application areas, ranging from register allocation, frequency association in telecommunications, timetabling and scheduling, and others.
A. F. Silva +2 more
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Canadian Journal of Mathematics, 1961
A graph on n labelled nodes is a set of n objects called “nodes”, distinguishable from each other, and a set (possibly empty) of “edges,” that is, pairs of nodes. Each edge is said to join its pair of nodes, at most one edge joins any two nodes and no edge joins a node to itself.
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A graph on n labelled nodes is a set of n objects called “nodes”, distinguishable from each other, and a set (possibly empty) of “edges,” that is, pairs of nodes. Each edge is said to join its pair of nodes, at most one edge joins any two nodes and no edge joins a node to itself.
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Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gröbner Bases
Cybersecurity and Cyberforensics Conference, 2017We consider the graph k-colouring problem encoded as a set of polynomial equations in the standard way. We prove that there are bounded-degree graphs that do not have legal k-colourings but for which the polynomial calculus proof system defined in [Clegg
Massimo Lauria, Jakob Nordström
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Colouring a graph with position sets
Ars Mathematica ContemporaneaIn this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour the vertices of the graph such that each colour class has the no-three-in-line property ...
Ullas Chandran S.V. +4 more
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Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth
Combinatorics, Probability and Computing, 1998For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G ...
Emden-Weinert, Thomas +2 more
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Bulletin of the London Mathematical Society, 1989
We prove that the TCC (Total Colouring Conjecture) is true for complete r-partite graphs which extends a result of M. Rosenfeld. We also give an alternate, slightly simpler proof of an earlier result (which says that the TCC is true for graphs having maximum degree 3) obtained independently by M. Rosenfeld and N. Vijayaditya.
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We prove that the TCC (Total Colouring Conjecture) is true for complete r-partite graphs which extends a result of M. Rosenfeld. We also give an alternate, slightly simpler proof of an earlier result (which says that the TCC is true for graphs having maximum degree 3) obtained independently by M. Rosenfeld and N. Vijayaditya.
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Unsolved graph colouring problems
2015Our book Graph Coloring Problems [85] appeared in 1995. It contains descriptions of unsolved problems, organized into sixteen chapters. A large number of publications on graph colouring have appeared since then, and in particular around thirty of the 211 problems in that book have been solved.
Jensen, Tommy, Toft, Bjarne
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Information Processing Letters, 2000
The problem we are addressing in this paper was proposed at SIROCCO'98 by Kranakis, with possible applications to finding a consensus in distributive networks. The problem can be viewed as a simple game on graphs, in a way similar to the game of life.
Stefan Dobrev +3 more
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The problem we are addressing in this paper was proposed at SIROCCO'98 by Kranakis, with possible applications to finding a consensus in distributive networks. The problem can be viewed as a simple game on graphs, in a way similar to the game of life.
Stefan Dobrev +3 more
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Algorithms for Colouring Random k-colourable Graphs
Combinatorics, Probability and Computing, 2000Random \(k\)-colourable graphs on \(n\) vertices are obtained by partitioning the vertices into \(k\) colour classes and selecting edges independently with a common probability for all pairs of vertices from different colour classes. The colour classes are either fixed or random.
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