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Acyclic Colorings of Graph Subdivisions [chapter]

Debajyoti Mondal, Rahnuma Islam Nishat, Sue Whitesides, Md. Saidur Rahman
2011 Lecture Notes in Computer Science  
Furthermore, we give some sufficient conditions on the number 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  ... 
doi:10.1007/978-3-642-25011-8_20 fatcat:qiem4jshorfftp6f2s3npzvsuu

Acyclic colorings of graph subdivisions revisited

Debajyoti Mondal, Rahnuma Islam Nishat, Sue Whitesides, Md. Saidur Rahman
2012 Journal of Discrete Algorithms  
Furthermore, we give some upper bounds on the number 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.  ... 
doi:10.1016/j.jda.2012.06.001 fatcat:e6qycgbmgran3ljkgut5o54hom

Acyclically 3-colorable planar graphs

Patrizio Angelini, Fabrizio Frati
2011 Journal of combinatorial optimization  
Further, we show that every planar 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  ... 
doi:10.1007/s10878-011-9385-3 fatcat:jurhgxylzjajpj6nr4menlfgiq

Acyclic Coloring with Few Division Vertices [chapter]

Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman, Sue Whitesides
2012 Lecture Notes in Computer Science  
In this paper we prove that every triangulated plane 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.  ... 
doi:10.1007/978-3-642-35926-2_11 fatcat:b6kdafg26zajzexo3qr45ms5ge

Acyclic coloring with few division vertices

Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman, Sue Whitesides
2013 Journal of Discrete Algorithms  
In this paper we prove that every triangulated plane 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] .  ... 
doi:10.1016/j.jda.2013.08.002 fatcat:t465grxbofhl7hcik22p7kefry

Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices

Federico Ardila, Cesar Ceballos
2013 Discrete & Computational Geometry  
A fine mixed 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  ... 
doi:10.1007/s00454-013-9485-1 fatcat:76uxeo5ucrgldfrbmtbezkufmi

On forbidden subdivision characterizations of graph classes

Zdeněk Dvořák
2008 European journal of combinatorics (Print)  
We provide a characterization 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).  ... 
doi:10.1016/j.ejc.2007.05.008 fatcat:lwufzyuoibehlkn7vaaxhos6ia

Dichromatic number and forced subdivisions [article]

Lior Gishboliner, Raphael Steiner, Tibor Szabó
2020 arXiv   pre-print
This is an extension 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  ... 
arXiv:2008.09888v1 fatcat:a4eh4memerfpncn6ff473xpe2a

Fraternal augmentations, arrangeability and linear Ramsey numbers

Jaroslav Nešetřil, Patrice Ossona de Mendez
2009 European journal of combinatorics (Print)  
This implies the linearity 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.  ... 
doi:10.1016/j.ejc.2009.03.012 fatcat:brgbj4qwobhgno7f7uzjhecfou

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.  ... 

Chordal Graphs are Fully Orientable [article]

Hsin-Hao Lai, Ko-Wei Lih
2012 arXiv   pre-print
Suppose that D is an 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.  ... 
arXiv:1202.5718v1 fatcat:76hgvjrbyvgcnc5jrjua3onlfi

Bounds on Edge Colorings with Restrictions on the Union of Color Classes

N. R. Aravind, C. R. Subramanian
2010 SIAM Journal on Discrete Mathematics  
This generalizes some well-known types 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  ... 
doi:10.1137/080733917 fatcat:klpb3bhhwvdgdn5q3vrlqmq4am

Complete minors in digraphs with given dichromatic number [article]

Tamás Mészáros, Raphael Steiner
2021 arXiv   pre-print
The dichromatic number χ⃗(D) 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.  ... 
arXiv:2101.04590v1 fatcat:2zgl4z4covbqpoea5lvokfnqsm

Acyclically pushable bipartite permutation digraphs: An algorithm

Romeo Rizzi
2006 Discrete Mathematics  
and also provides linear time algorithms to find a strongly 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.  ... 
doi:10.1016/j.disc.2005.11.027 fatcat:44s3oorjbjb6zmcdw3tfbx2ksu

On the Total Chromatic Edge Stability Number and the Total Chromatic Subdivision Number of Graphs

2022 Discrete Mathematics Letters  
(ii) The total chromatic 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.  ... 
doi:10.47443/dml.2021.111 fatcat:62m5o74gnrhdzdowcl4443ieyq
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