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Conflict-Free Colouring of Subsets [article]

Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, Yelena Yuditsky
2022 arXiv   pre-print
We introduce and study conflict-free colourings of t-subsets in hypergraphs.  ...  (iii) We also show that there is no general bound on the t-subset conflict-free chromatic number as a function of the standard conflict-free chromatic number already for t=2.  ...  Conflict-Free Colouring of Subsets  ... 
arXiv:2207.01041v1 fatcat:ysbha4zjefhd7efm2rffqf3opq

Perfect Resolution of Strong Conflict-Free Colouring of Interval Hypergraphs [article]

S.M. Dhannya, N.S. Narayanaswamy
2021 arXiv   pre-print
The k-Strong Conflict-Free (k-SCF, in short) colouring problem seeks to find a colouring of the vertices of a hypergraph H using minimum number of colours so that in every hyperedge e of H, there are at  ...  least min{|e|,k} vertices whose colour is different from that of all other vertices in e.  ...  I ℓ } based on t and the vertex colouring as follows: the set I i consists of all those hyperedges I ∈ I such that the colour of t(I) is i.  ... 
arXiv:1707.05071v4 fatcat:qi2cfpqrwvahdjjuvlx264tvxe

Conflict-Free Colourings of Graphs and Hypergraphs

JÁNOS PACH, GÁBOR TARDOS
2009 Combinatorics, probability & computing  
A colouring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of 'unique' colour that does not get repeated in E.  ...  The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by χ CF (H).  ...  Conflict-Free Colourings of Graphs and Hypergraphs  ... 
doi:10.1017/s0963548309990290 fatcat:47cm2tiqqvefdp4jdq7phjjx3q

Conflict-free vertex connection number at most 3 and size of graphs

Trung Duy Doan, Ingo Schiermeyer
2019 Discussiones Mathematicae Graph Theory  
A path in a vertex-coloured graph is called conflict-free if there is a colour used on exactly one of its vertices.  ...  A vertex-coloured graph is said to be conflict-free vertex-connected if any two distinct vertices of the graph are connected by a conflict-free vertex-path.  ...  The conflict-free connection number of a connected graph G, denoted by cf c(G), is defined as the smallest number of colours in order to make it conflict-free connected.  ... 
doi:10.7151/dmgt.2211 fatcat:5unwwpaxxrdjrcmxvqlr7ipmay

On conflict-free chromatic guarding of simple polygons [article]

Onur Çağırıcı and Subir Kumar Ghosh and Petr Hliněný and Bodhayan Roy
2020 arXiv   pre-print
This is also known as the conflict-free chromatic guarding problem with vertex guards, and is motivated, e.g., by the problem of radio frequency assignment to sensors placed at the polygon vertices.  ...  We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used.  ...  We recall the following property from the proof of Proposition 6.1: (i) In any conflict-free 2-colouring of the bowl there is a guard (or two) placed at the door.  ... 
arXiv:1904.08624v3 fatcat:i5kh5en7src2zhdr7ulic5gcc4

Invariant measures in simple and in small theories [article]

Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupinski, Slavko Moconja, Anand Pillay, Nicholas Ramsey
2022 arXiv   pre-print
groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.  ...  We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable  ...  Colourings and free actions. We fix a free action of F 5 on a set X. As above, we will denote by {f 1 , f 2 , f 3 , g 1 , g 2 } a system of free generators for the free group F 5 .  ... 
arXiv:2105.07281v2 fatcat:iekcizjczvcbva5tbxliws45s4

Linear combinations of heuristics for examination timetabling

Edmund K. Burke, Nam Pham, Rong Qu, Jay Yellen
2011 Annals of Operations Research  
Some recent work has demonstrated the superiority of combining a number of different heuristics for vertex and colour selection.  ...  Most of this work used a single heuristic to measure the difficulty of a vertex. Relatively less attention has been paid to select an appropriate colour for the selected vertex.  ...  The weight of an edge between two vertices represents the number of students taking both corresponding exams. Conflict-free colourings represent timetables that satisfy the hard constraint.  ... 
doi:10.1007/s10479-011-0854-y fatcat:2jf7dk7s2rcpjkipdiwd7y7gni

Perfect Resolution of Conflict-Free Colouring of Interval Hypergraphs

S. M. Dhannya, N. S. Narayanaswamy, Markus Bläser, Christophe Paul
2020 Symposium on Theoretical Aspects of Computer Science  
Given a hypergraph H, the conflict-free colouring problem is to colour vertices of H using minimum colours so that in every hyperedge e of H, there is a vertex whose colour is different from that of all  ...  Our results are on a variant of the conflict-free colouring problem considered by Cheilaris et al.  ...  [4] and is a variant of a well-studied hypergraph colouring problem known as the Conflict-Free colouring problem.  ... 
doi:10.4230/lipics.stacs.2020.52 dblp:conf/stacs/DhannyaN20 fatcat:gukkufd6afhv3pwuzccy5ksnyu

A Lower Bound on the Area of a 3-Coloured Disc Packing

Peter Brass, Ferran Hurtado, Benjamin J. Lafreniere, Anna Lubiw
2007 Canadian Conference on Computational Geometry  
Motivated by the problem of channel-assignment for wireless access points, we consider a variant where the selected subset of discs must be 3-colourable, with discs of the same colour pairwisedisjoint.  ...  Given a set of unit-discs in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the discs? Rado conjectured 1/4 and proved 1/4.41.  ...  There is also work on online algorithms for conflict-free colouring [8] , and on conflictfree colouring of regions other than discs [10] .  ... 
dblp:conf/cccg/BrassHLL07 fatcat:dwo4ylxgk5av3mncrpk7czoanm

A Lower Bound on the Area of a 3-Coloured Disk Packing [article]

Peter Brass, Ferran Hurtado, Benjamin Lafreniere, Anna Lubiw
2008 arXiv   pre-print
Motivated by the problem of channel-assignment for wireless access points, in which use of 3 channels is a standard practice, we consider a variant where the selected subset of disks must be 3-colourable  ...  with disks of the same colour pairwise-disjoint.  ...  Keshav, University of Waterloo, and the work was initiated at the 5th  ... 
arXiv:0804.1173v1 fatcat:cw6trtnlovagpmqv6wuxjhebxi

An Optimal Algorithm for Conflict-Free Coloring for Tree of Rings [article]

Einollah Pira
2012 arXiv   pre-print
An optimal algorithm is presented about Conflict-Free Coloring for connected subgraphs of tree of rings. Suppose the number of the rings in the tree is |T| and the maximum length of rings is |R|.  ...  A presented algorithm in [1] for a Tree of rings used O(log|T|.log|R|) colors but this algorithm uses O(log|T|+log|R|) colors.  ...  ACKNOWLEDGEMENTS This research has been supported by the business training center of tabriz,Iran.  ... 
arXiv:1203.2414v1 fatcat:wsfhx4rcvrhqlhcdt2ogtu7id4

Colouring graphs with no induced six-vertex path or diamond [article]

Jan Goedgebeur, Shenwei Huang, Yiao Ju, Owen Merkel
2021 arXiv   pre-print
In this paper we show that the chromatic number of a (P_6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G.  ...  We do this by reducing the problem to imperfect (P_6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each  ...  By (12) , B and C do not conflict. By (11) (13) , this is a proper colouring of F . By (18) , F and C do not conflict. By (17) , B 2,3 and F \ F 5 do not conflict.  ... 
arXiv:2106.08602v1 fatcat:pl3t44exu5elxorxxjb43hhl3m

Are scale-free regulatory networks larger than random ones? [article]

Miguel A. Fortuna, Carlos J. Melian
2005 arXiv   pre-print
Network of packages with regulatory interactions (dependences and conflicts) from Debian GNU/Linux operating system is compiled and used as analogy of a gene regulatory network.  ...  Using a trace-back algorithm we assembly networks from the potential pool of packages for both scale-free and exponential topology from real and a null model data, respectively.  ...  Colours in the horizontal bars correspond to the number of each type of package in the null model. Yellow are packages with k con in > 0 or k con out > 0.  ... 
arXiv:q-bio/0501027v1 fatcat:cgg2vxjtpvfn5jjoxevi633xwu

An Optimal Algorithm for Conflict-Free Coloring for Tree of Rings

Einollah Pira
2012 Computer Science & Engineering An International Journal  
An optimal algorithm is presented about Conflict-Free Coloring for connected subgraphs of tree of rings. Suppose the number of the rings in the tree is |T| and the maximum length of rings is |R|.  ...  A presented algorithm in [1] for a Tree of rings used O(log|T|.log|R|) colors but this algorithm uses O(log|T|+log|R|) colors.  ...  ACKNOWLEDGEMENTS This research has been supported by the business training center of tabriz,Iran.  ... 
doi:10.5121/cseij.2012.2101 fatcat:g66n3ftkunfhtfxde5asul5e2u

Improved lower bounds on parity vertex colourings of binary trees [article]

Jan Soukup
2019 arXiv   pre-print
These lower bounds are also lower bounds for the conflict-free colouring. We also prove that χ_p(G) is not monotone with respect to minors and determine its value for cycles.  ...  A vertex colouring is called a parity vertex colouring if every path in G contains an odd number of occurrences of some colour.  ...  The conflict-free colouring is a relaxation of a unique-maximum colouring.  ... 
arXiv:1910.03341v2 fatcat:op2bgvoa4jbyvp5bvwuvnz2n4u
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