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Conflict-free coloring of graphs [article]

Roman Glebov, Tibor Szabó, Gábor Tardos
2013 arXiv   pre-print
We study the conflict-free chromatic number chi_CF of graphs from extremal and probabilistic point of view.  ...  In relation to this we study the evolution of the conflict-free chromatic number of the Erdős-Rényi random graph G(n,p) and give the asymptotics for p=omega(1/n).  ...  We would like to thank Yury Person for pointing out an error in an earlier version of the proof of Theorem 4.  ... 
arXiv:1111.5501v2 fatcat:uq3dsoqp25fszma7blbhay35ie

Conflict-Free Coloring of Graphs

Zachary Abel, Victor Alvarez, Erik D. Demaine, Sándor P. Fekete, Aman Gour, Adam Hesterberg, Phillip Keldenich, Christian Scheffer
2018 SIAM Journal on Discrete Mathematics  
conflict-free coloring of G.  ...  Conflict-free coloring has received an increasing amount of attention. Because of the relationship to classic coloring, it is natural to investigate the conflict-free coloring of planar graphs.  ... 
doi:10.1137/17m1146579 fatcat:uj3b7rzgsbgb3io4j7njd2i6dy

Conflict-Free Coloring of String Graphs [article]

Chaya Keller and Alexandre Rok and Shakhar Smorodinsky
2019 arXiv   pre-print
Conflict-free coloring (in short, CF-coloring) of a graph G = (V,E) is a coloring of V such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex  ...  ., intersection graphs of frames) we prove a tight bound of Θ( n) with respect to the notion of k-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most k  ...  Introduction Background Conflict-free coloring and k-conflict-free coloring of hypergraphs. Definition 1.1. Let H = (V, E) be a hypergraph and let c be a coloring c : V → N.  ... 
arXiv:1712.04524v2 fatcat:mdd7xxo2azew3cs74sku4rripy

Conflict-Free Coloring of Intersection Graphs [article]

Sándor P. Fekete, Phillip Keldenich
2017 arXiv   pre-print
A conflict-free k-coloring of a graph G=(V,E) assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and  ...  Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs.  ...  Unit square graphs of height 2 are conflict-free 2-colorable. Unit square graphs are conflict-free 4-colorable.  ... 
arXiv:1709.03876v1 fatcat:ssd4liw3sncb5peerltvokuora

Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs [article]

Sriram Bhyravarapu, Tim A. Hartmann, Hung P. Hoang, Subrahmanyam Kalyanasundaram, I. Vinod Reddy
2022 arXiv   pre-print
Given an undirected graph G, a conflict-free coloring CFON* (resp.  ...  The conflict-free coloring problem asks to find the minimum number of colors required for such a CFON* (resp.  ...  We would also like to thank Alexander Hermans for his help on finding a lower bound example for interval graphs.  ... 
arXiv:2105.08693v3 fatcat:e5d7n67opfgvlorwvownsaa5xe

Proper conflict-free coloring of sparse graphs [article]

Eun-Kyung Cho, Ilkyoo Choi, Hyemin Kwon, Boram Park
2022 arXiv   pre-print
A proper conflict-free c-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood.  ...  ., who proved that planar graphs have a proper conflict-free 8-coloring and constructed a planar graph with no proper conflict-free 5-coloring.  ...  It is not hard to see that a graph G with mad(G) ≤ c − 1 has a proper conflict-free c-coloring for c ∈ {1, 2}.  ... 
arXiv:2203.16390v2 fatcat:cxyfunoklvgnbfa3uj6x7x52ae

Conflict-free coloring on closed neighborhoods of bounded degree graphs [article]

Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew
2020 arXiv   pre-print
The closed neighborhood conflict-free chromatic number of a graph G, denoted by χ_CN(G), is the minimum number of colors required to color the vertices of G such that for every vertex, there is a color  ...  Comput. 2014], Glebov, Szabó and Tardos showed existence of graphs G with χ_CN(G) = Ω(log^2Δ). In this paper, we bridge the gap between the two bounds by showing that χ_CN(G) = O(log^2 Δ).  ...  Conflict-free coloring has also been studied in the context of hypergraphs created out of simple graphs.  ... 
arXiv:2003.05637v3 fatcat:2fvvv3pcwbgdbgmqwbxijviqbe

Parameterized Complexity of Conflict-free Graph Coloring [article]

Hans L. Bodlaender, Sudeshna Kolay, Astrid Pieterse
2019 arXiv   pre-print
Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring c:V(G) →{1,2,...  ...  When we replace N(v) by the closed neighborhood N[v], then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring.  ...  The research was partially done when the first and second author were associated with Eindhoven University of Technology.  ... 
arXiv:1905.00305v1 fatcat:wpmqja7okjcmtkoszvdfvzcam4

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Zachary Abel, Victor Alvarez, Erik D. Demaine, Sándor P. Fekete, Aman Gour, Adam Hesterberg, Phillip Keldenich, Christian Scheffer
2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms  
a planar bipartite graph has a conflict-free k-coloring.  ...  A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors  ...  Work on this paper was partially supported by the DFG Research Unit "Controlling Concurrent Change", funding number FOR 1800, project FE407/17-2, "Conflict Resolution and Optimization".  ... 
doi:10.1137/1.9781611974782.127 dblp:conf/soda/AbelADFGHKS17 fatcat:swnroxpghrgcbjwzixamj3gb5i

Parameterized Algorithms for Conflict-free Colorings of Graphs [article]

I. Vinod Reddy
2017 arXiv   pre-print
The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free.  ...  In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood.  ...  The minimum number of colors needed to conflict-free color the vertices of a hypergraph H is called the conflict-free chromatic number of H.  ... 
arXiv:1710.00223v1 fatcat:3pecnfkahzbdhdqbrco2vplpwe

Conflict-free coloring on open neighborhoods of claw-free graphs [article]

Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew
2021 arXiv   pre-print
The 'Conflict-Free Open (Closed) Neighborhood coloring', abbreviated CFON (CFCN) coloring, of a graph G using r colors is a coloring of the vertices of G such that every vertex sees some color exactly  ...  As an open question, they asked if the result could be extended to claw-free (K_1,3-free) graphs, which are a superclass of line graphs.  ...  Conflict-free chromatic number versus conflict-free chromatic index. Journal of Graph Theory, 2021. [EL75] P. Erdős and L. Lovász.  ... 
arXiv:2112.12173v1 fatcat:ddqsqaqucrftloo2lquegzbiqa

Conflict-Free Coloring of Star-Free Graphs on Open Neighborhoods [article]

Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew
2020 arXiv   pre-print
Given a graph, the conflict-free coloring problem on open neighborhoods (CFON) asks to color the vertices of the graph so that all the vertices have a uniquely colored vertex in its open neighborhood.  ...  The smallest number of colors required for such a coloring is called the conflict-free chromatic number and denoted χ_ON(G).  ...  to Definition 1) of line graphs is O(log ∆).  ... 
arXiv:2009.06720v1 fatcat:2zzmslartfbkfpnthilhdf5ccq

Conflict-Free Coloring of Intersection Graphs of Geometric Objects [article]

Chaya Keller, Shakhar Smorodinsky
2017 arXiv   pre-print
Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of \rho-fat  ...  A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices  ...  Dozens of papers studied conflict-free colorings of graphs and hypergraphs, mostly arising from geometric objects. See the survey [Smo13] and the references therein for more on CF-colorings.  ... 
arXiv:1704.02018v1 fatcat:i7l6nnvd6febjje67gbpt6pwjy

Graph unique-maximum and conflict-free colorings

Panagiotis Cheilaris, Géza Tóth
2011 Journal of Discrete Algorithms  
In a conflict-free coloring, in every path of the graph there is a color that appears only once.  ...  We investigate the relationship between two kinds of vertex colorings of graphs: uniquemaximum colorings and conflict-free colorings.  ...  Conflict-free coloring of graphs with respect to paths is a special case of conflict-free colorings of hypergraphs, first studied in [7, 17] .  ... 
doi:10.1016/j.jda.2011.03.005 fatcat:cxnfwcqytfgejfbw3cao75jxye

FPT Algorithms for Conflict-free Coloring of Graphs and Chromatic Terrain Guarding [article]

Akanksha Agrawal, Pradeesha Ashok, Meghana M Reddy, Saket Saurabh, Dolly Yadav
2019 arXiv   pre-print
We also initiate the study of Strong Conflict-free Coloring of graphs.  ...  Given a graph G=(V,E), conflict-free coloring of G refers to coloring a subset of V such that for every vertex v, there is a color that is assigned to exactly one vertex in the closed neighborhood of v  ...  We color v with color 1 . The convex layer is now divided into two halves.  ... 
arXiv:1905.01822v1 fatcat:7kqc2ybmzfgsliti72rtntko2y
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