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Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

Hajo Broersma, Fedor V. Fomin, Jan Kratochvil, Gerhard J. Woeginger
2005 Algorithmica  
2006) 'Planar graph coloring avoiding monochromatic subgraphs : trees and paths make it difficult.', Algorithmica., 44 (4). pp. 343-361.  ...  Abstract We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs.  ...  Acknowledgments We are grateful to Oleg Borodin, Alesha Glebov, Sasha Kostochka, and Carsten Thomassen for fruitful discussions on the topic of this paper.  ... 
doi:10.1007/s00453-005-1176-8 fatcat:feseqsxyr5h5ddrp3kqsvk6bxu

On-line Ramsey Theory for Bounded Degree Graphs

Jane Butterfield, Tracy Grauman, William B. Kinnersley, Kevin G. Milans, Christopher Stocker, Douglas B. West
2011 Electronic Journal of Combinatorics  
In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class ${\cal H}$.  ...  When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph $G$.  ...  We note a simpler proof of the weaker boundR ∆ (G) ≤ 8 when ∆(G) ≤ 2; this proof also avoids Lemma 5.5. The case of even b is as before.  ... 
doi:10.37236/623 fatcat:rx6c6374j5h4rgwhtnaqy6pa6m

On the complexity of bicoloring clique hypergraphs of graphs

J KRATOCHVIL, Z TUZA
2002 Journal of Algorithms  
., whether the vertices of G can be colored with two colors so that no maximal clique is monochromatic.  ...  Given a graph G, its clique hypergraph C(G) has the same set of vertices as G and the hyperedges correspond to the (inclusionwise) maximal cliques of G.  ...  In the graph G obtained, every 2-coloring of the cliques makes the neighborhood of x monochromatic. Therefore, C(G) is 2colorable if and only if so is the original hypergraph H.  ... 
doi:10.1016/s0196-6774(02)00221-3 fatcat:wv3mfkprdfggrnqrbvhe5g7csm

Defective and Clustered Graph Colouring [article]

David R. Wood
2018 arXiv   pre-print
The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a  ...  graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding K_t as a minor, graphs excluding K_s,t as a minor, and graphs excluding an arbitrary graph  ...  and clustered colourings.  ... 
arXiv:1803.07694v1 fatcat:prcytiquyzb77cuxaexdflx5lm

Vertex-Coloring with Star-Defects [article]

Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Vincenzo Roselli
2016 arXiv   pre-print
Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors.  ...  Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic  ...  In particular, we call a graph G tree-diameter-λ κ-colorable if the vertices of G can be colored with κ colors, so that all monochromatic components are acyclic and of diameter at most λ, where κ ≥ 1,  ... 
arXiv:1512.02505v2 fatcat:vahygnrwcfbbtpyntmsu3dciye

Exact defective colorings of graphs [article]

James Cumberbatch, Juho Lauri, Christodoulos Mitillos
2021 arXiv   pre-print
We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs.  ...  An exact (k,d)-coloring of a graph G is a coloring of its vertices with k colors such that each vertex v is adjacent to exactly d vertices having the same color as v.  ...  It is not difficult to see that for particular values of k and d, it can be that a graph G does not admit an exact (k, d)-coloring.  ... 
arXiv:2109.05255v1 fatcat:bzge2rr4srcdjhbp3lq2sz4jma

Eulerian Circuits with No Monochromatic Transitions in Edge-Colored Digraphs with all Vertices of Outdegree Three

James M. Carraher, Stephen G. Hartke
2017 SIAM Journal on Discrete Mathematics  
A colored eulerian digraph is an eulerian digraph G where a color is assigned to the tail of each edge and a color is assigned to the head of each edge.  ...  Let S 3 be the set of vertices of outdegree and indegree three that have exactly three colors on the incident edges where each color appears on exactly one incoming and exactly one outgoing edge.  ...  Acknowledgements We thank the referees for their helpful comments and suggestions.  ... 
doi:10.1137/140992850 fatcat:6lefbpeu3zct3avrxi7povjnzy

From the plane to higher surfaces

Ken-ichi Kawarabayashi, Carsten Thomassen
2012 Journal of combinatorial theory. Series B (Print)  
These include the 5-list-color theorem, results on arboricity, and various types of colorings, and decomposition theorems of planar graphs into two graphs with prescribed degeneracy properties.  ...  It is not known if the 4-color theorem extends in this way.  ...  It is not difficult to prove that every planar graph is 5-colorable. The Four Color Theorem [4, 19] shows that every planar graphs is 4-colorable.  ... 
doi:10.1016/j.jctb.2012.03.001 fatcat:kvckd6thxre7nnvtt3tf5vdtai

Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs

Pinar Heggernes, Davis Issac, Juho Lauri, Paloma T. Lima, Erik Jan Van Leeuwen, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors.  ...  We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices.  ...  A graph is a diametral path if every connected induced subgraph has a dominating diametral path.  ... 
doi:10.4230/lipics.mfcs.2018.83 dblp:conf/mfcs/HeggernesILLL18 fatcat:3zrivhwysbgxrliwami56jyynu

Visual Analytics for Sets over Time and Space (Dagstuhl Seminar 19192)

Sara Irina Fabrikant, Silvia Miksch, Alexander Wolff, Michael Wagner
2019 Dagstuhl Reports  
point of view (graph drawing, computational geometry, and cognition), (ii) from a temporal point of view (visual analytics and information visualization over time, HCI), and (iii) from a space-time point  ...  The goal of the seminar was to identify specific theoretical and practical problems that need to be solved in order to create dynamic and interactive set visualizations that take into account time and  ...  In particular, it was great to have the opportunity to use a separate room for each working group.  ... 
doi:10.4230/dagrep.9.5.31 dblp:journals/dagstuhl-reports/FabrikantM019 fatcat:ng3ajyjdbrhdhmb7xiiigohsde

Decomposition of cubic graphs related to Wegner's conjecture [article]

János Barát
2019 arXiv   pre-print
Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree 1 (that is, it consists of a matching and some  ...  isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path.  ...  Therefore, it seems difficult to make this simple idea into a full proof.  ... 
arXiv:1901.11339v1 fatcat:sxzvgsl7xzbi5kynbq3tf6ylzy

The Complexity of Bottleneck Labeled Graph Problems

Refael Hassin, Jérôme Monnot, Danny Segev
2008 Algorithmica  
The generic objective is to construct a subgraph of prescribed structure (such as that of being an s-t path, a spanning tree, or a perfect matching) while trying to avoid over-picking or under-picking  ...  We present hardness results, approximation heuristics, and exact algorithms for bottleneck labeled optimization problems arising in the context of graph theory.  ...  It is not difficult to verify that UL-max-min s-t path generalizes the longest path problem, even in monochromatic graphs.  ... 
doi:10.1007/s00453-008-9261-4 fatcat:vdv7wuagk5aqllzzb2ha5y6fri

The Graph Minor Algorithm with Parity Conditions

Ken-ichi Kawarabayashi, Bruce Reed, Paul Wollan
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
Although the original Graph Minor algorithm of Robertson and Seymour does depend on it and our proof does have similarities to their arguments, we can avoid the structure theorem by building on the shorter  ...  Here, however, we must maintain the parity of the paths and can only use an "odd clique minor". This requires new techniques to describe the structure of the graph when we cannot find such a minor.  ...  Moreover, this huge planar subgraph itself has very large tree-width. This makes it possible to prove that the "middle" vertex v of this planar subgraph is irrelevant.  ... 
doi:10.1109/focs.2011.52 dblp:conf/focs/KawarabayashiRW11 fatcat:sdteaextuffbpgcm722tajuzxy

Exact Algorithms for Coloring Graphs While Avoiding Monochromatic Cycles [chapter]

Fabrice Talla Nobibon, Cor Hurkens, Roel Leus, Frits C. R. Spieksma
2010 Lecture Notes in Computer Science  
of vertices that can be colored using two colors while avoiding monochromatic cycles.  ...  We consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs.  ...  avoiding monochromatic cycles.  ... 
doi:10.1007/978-3-642-14355-7_24 fatcat:r6gqayhmj5dtzcum37ra4jtkbq

Coloring Graphs Using Two Colors While Avoiding Monochromatic Cycles

Fabrice Talla Nobibon, Cor A. J. Hurkens, Roel Leus, Frits C. R. Spieksma
2012 INFORMS journal on computing  
of vertices that can be colored using two colors while avoiding monochromatic cycles.  ...  W e consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs.  ...  Acknowledgments The authors thank the referees and the associate editor for their constructive comments, and in particular for suggesting the alternative branching strategy.  ... 
doi:10.1287/ijoc.1110.0466 fatcat:k72723y4hfdl7dfofwfcgvevpu
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