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Acyclic Colorings of Graph Subdivisions
[chapter]

2011
*
Lecture Notes in Computer Science
*

Furthermore, we give some sufficient conditions on the number

doi:10.1007/978-3-642-25011-8_20
fatcat:qiem4jshorfftp6f2s3npzvsuu
*of*division vertices for*acyclic*3-*coloring**of**subdivisions**of*partial k-trees and cubic*graphs*. ... An*acyclic**coloring**of*a*graph*G is a*coloring**of*the vertices*of*G, where no two adjacent vertices*of*G receive the same*color*and no cycle*of*G is bichromatic. ... Main Results : We study*acyclic**colorings**of**subdivisions**of**graphs*and prove the following claims. (1) Every cubic*graph*with n vertices has a*subdivision*that is*acyclically*3colorable, where the number ...##
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Acyclic colorings of graph subdivisions revisited

2012
*
Journal of Discrete Algorithms
*

Furthermore, we give some upper bounds on the number

doi:10.1016/j.jda.2012.06.001
fatcat:e6qycgbmgran3ljkgut5o54hom
*of*division vertices for*acyclic*3-*coloring**of**subdivisions**of*k-trees and cubic*graphs*. ... An*acyclic**coloring**of*a*graph*G is a*coloring**of*the vertices*of*G, where no two adjacent vertices*of*G receive the same*color*and no cycle*of*G is bichromatic. ... Acknowledgement The authors wish to thank the anonymous referees for bringing their attention to an error in Theorem 8 in an initial submission*of*the paper. ...##
###
Acyclically 3-colorable planar graphs

2011
*
Journal of combinatorial optimization
*

Further, we show that every planar

doi:10.1007/s10878-011-9385-3
fatcat:jurhgxylzjajpj6nr4menlfgiq
*graph*has a*subdivision*with one vertex per edge that admits an*acyclic*3-*coloring*. ... We show that testing*acyclic*3-*colorability*is N P-hard, even for planar*graphs**of*maximum degree 4, and we show that there exist infinite classes*of*cubic planar*graphs*that are not*acyclically*3-*colorable*...*Acyclic**colorings**of**graph**subdivisions*have been already considered by Wood in [17] , where the author observed that every*graph*has a*subdivision*with two vertices per edge that is*acyclically*3-*colorable*...##
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Acyclic Coloring with Few Division Vertices
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this paper we prove that every triangulated plane

doi:10.1007/978-3-642-35926-2_11
fatcat:b6kdafg26zajzexo3qr45ms5ge
*graph*with n vertices has a 1-*subdivision*that is*acyclically*3-*colorable*(respectively, 4-*colorable*), where the number*of*division vertices is at most ... An*acyclic*k-*coloring**of*a*graph*G is a mapping φ from the set*of*vertices*of*G to a set*of*k distinct*colors*such that no two adjacent vertices receive the same*color*and φ does not contain any bichromatic ... Acknowledgment We thank Bangladesh Academy*of*Sciences (BAS) for providing research travel grants to Md. Saidur Rahman for presenting the paper at IWOCA 2012. ...##
###
Acyclic coloring with few division vertices

2013
*
Journal of Discrete Algorithms
*

In this paper we prove that every triangulated plane

doi:10.1016/j.jda.2013.08.002
fatcat:t465grxbofhl7hcik22p7kefry
*graph*with n vertices has a 1-*subdivision*that is*acyclically*3-*colorable*(respectively, 4-*colorable*), where the number*of*division vertices is at most ... An*acyclic*k-*coloring**of*a*graph*G is a mapping φ from the set*of*vertices*of*G to a set*of*k distinct*colors*such that no two adjacent vertices receive the same*color*and φ does not contain any bichromatic ...*Acyclic**colorings**of**graphs*and their*subdivisions*find applications in diverse areas [16] [17] [18] . ...##
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Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices

2013
*
Discrete & Computational Geometry
*

A fine mixed

doi:10.1007/s00454-013-9485-1
fatcat:76uxeo5ucrgldfrbmtbezkufmi
*subdivision**of*a (d-1)-simplex T*of*size n gives rise to a system*of*d 2 permutations*of*[n] on the edges*of*T, and to a collection*of*n unit (d-1)-simplices inside T. ... We propose and give evidence for an answer to the first question, the*Acyclic*System Conjecture. We prove that the system*of*permutations*of*T determines the collection*of*simplices*of*T. ... Acknowledgments We would like to thank Hwanchul Yoo for his example in Sect. 8.1, and Christopher O'Neill for running some computational experiments in support*of*the Weak Spread Out Simplices Conjecture ...##
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On forbidden subdivision characterizations of graph classes

2008
*
European journal of combinatorics (Print)
*

We provide a characterization

doi:10.1016/j.ejc.2007.05.008
fatcat:lwufzyuoibehlkn7vaaxhos6ia
*of*several*graph*parameters (the*acyclic*chromatic number, the arrangeability, and a sequence*of*parameters related to the expansion*of*a*graph*) in terms*of*forbidden*subdivisions*... Acknowledgements I would like to thank Jaroslav Nešetřil for discussions that led me to consider this problem, and to Hal Kierstead for pointing out the possible relationship with the game*coloring*number ... Theorem 3 is quite large: The*graph*K n,n has*acyclic*chromatic number n+1, but each*graph*whose 1-*subdivision*is a subgraph*of*K n,n has chromatic number O( √ n). ...##
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Dichromatic number and forced subdivisions
[article]

2020
*
arXiv
*
pre-print

This is an extension

arXiv:2008.09888v1
fatcat:a4eh4memerfpncn6ff473xpe2a
*of*the classical result by Dirac that 4-chromatic*graphs*contain a K_4-*subdivision*to directed*graphs*. ... For a digraph F, denote by mader_χ⃗(F) the smallest integer k such that every k-dichromatic digraph contains a*subdivision**of*F. ... Acknowledgement The research on this project was initiated during a joint research workshop*of*Tel Aviv University and the Freie Universität Berlin on Ramsey Theory, held in Tel Aviv in March 2020, and ...##
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Fraternal augmentations, arrangeability and linear Ramsey numbers

2009
*
European journal of combinatorics (Print)
*

This implies the linearity

doi:10.1016/j.ejc.2009.03.012
fatcat:brgbj4qwobhgno7f7uzjhecfou
*of*the Ramsey number and the bounded game chromatic number for some new classes*of**graphs*. ... We relate the notions*of*arrangeability and admissibility to bounded expansion classes and prove that these notions can be characterized by ∇ 1 (G). (The Burr-Erdős conjecture relates to ∇ 0 (G).) ... It is not known whether the structure*of**graphs*with unbounded ∇ 1 and bounded ∇ 0 could be characterized. ...##
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Page 6926 of Mathematical Reviews Vol. , Issue 2001J
[page]

2001
*
Mathematical Reviews
*

In this paper we demonstrate the smallest HC

*graphs*for the best known*coloring*heuristics in classical applications, as well as when adapted to the chromatic sum*coloring*and strong*coloring**of*vertices ... Jagger, who proved a number*of*extremal results concerning*subdivisions**of*digraphs [European J. ...##
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Chordal Graphs are Fully Orientable
[article]

2012
*
arXiv
*
pre-print

Suppose that D is an

arXiv:1202.5718v1
fatcat:76hgvjrbyvgcnc5jrjua3onlfi
*acyclic*orientation*of*a*graph*G. An arc*of*D is called dependent if its reversal creates a directed cycle. ... Let m and M denote the minimum and the maximum*of*the number*of*dependent arcs over all*acyclic*orientations*of*G. ... A*subdivision**of*an edge*of*a*graph*is obtained by replacing that edge by a path consisting*of*new internal vertices. A*subdivision**of*a*graph*is obtained through a sequence*of**subdivisions**of*edges. ...##
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Bounds on Edge Colorings with Restrictions on the Union of Color Classes

2010
*
SIAM Journal on Discrete Mathematics
*

This generalizes some well-known types

doi:10.1137/080733917
fatcat:klpb3bhhwvdgdn5q3vrlqmq4am
*of**colorings*such as*acyclic*edge*colorings*, distance-2 edge*colorings*, low treewidth edge*colorings*, etc. ... We consider constrained proper edge*colorings**of*the following type: Given a positive integer j and a family F*of*connected*graphs*on 3 or more vertices, we require that the subgraph formed by the union ... In the case*of**acyclic**coloring*, it turns out (see [1] ) that any*acyclic*vertex*coloring*requires Ω(d 4/3 )*colors*for some*graphs*while, as mentioned before, an*acyclic*edge*coloring*is always possible ...##
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Complete minors in digraphs with given dichromatic number
[article]

2021
*
arXiv
*
pre-print

The dichromatic number χ⃗(D)

arXiv:2101.04590v1
fatcat:2zgl4z4covbqpoea5lvokfnqsm
*of*a digraph D is the smallest k for which it admits a k-*coloring*where every*color*class induces an*acyclic*subgraph. ... Inspired by Hadwiger's conjecture for undirected*graphs*, several groups*of*authors have recently studied the containment*of*directed*graph*minors in digraphs with given dichromatic number. ... Moreover, any extension*of*an*acyclic*{1, 2}-*coloring**of*D[X i0 ] to a {1, 2}-*coloring**of*D[X] where w 1 , . . . , w s−1 receive*color*1 and w s receives*color*2 is*acyclic*. ...##
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Acyclically pushable bipartite permutation digraphs: An algorithm

2006
*
Discrete Mathematics
*

and also provides linear time algorithms to find a strongly

doi:10.1016/j.disc.2005.11.027
fatcat:44s3oorjbjb6zmcdw3tfbx2ksu
*acyclic*orientation*of*an undirected*graph*, if one exists. ... We show how a result*of*Conforti et al [Balanced cycles and holes in bipartite*graphs*, Discrete Math. 199 (1-3) (1999) 27-33] can be essentially regarded as a characterization*of*strongly*acyclic*digraphs ... Nevertheless, the nodes*of*the intermediate*graphs*can all be regarded as nodes*of*G, hence we can assign them a*color*class and say whether they belong to U or V. ...##
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On the Total Chromatic Edge Stability Number and the Total Chromatic Subdivision Number of Graphs

2022
*
Discrete Mathematics Letters
*

(ii) The total chromatic

doi:10.47443/dml.2021.111
fatcat:62m5o74gnrhdzdowcl4443ieyq
*subdivision*number or χ -*subdivision*number sd χ (G) is the minimum number*of*edges*of*G whose*subdivision*results in a*graph*H ⊆ G with χ (H) = χ (G) or with E(H) = ∅. ... A proper total*coloring**of*a*graph*G is an assignment*of**colors*to the vertices and edges*of*G (together called the elements*of*G) such that neighbored elements-two adjacent vertices or two adjacent edges ... We also exactly determine es χ (G) for specific classes*of**graphs*such as*acyclic**graphs*, cycles, complete*graphs*, and complete bipartite*graphs*. ...
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