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Total colorings of planar graphs with large maximum degree

O. V. Borodin, A. V. Kostochka, D. R. Woodall
1997 Journal of Graph Theory  
It is proved that a planar graph with maximum degree ∆ ≥ 11 has total (vertex-edge) chromatic number ∆ + 1.  ...  If G is a planar graph with maximum degree ∆ ≥ 11, then χ (G) = ∆ + 1.  ...  With no condition on the girth, Borodin [2] proved it for planar graphs with maximum degree ∆ ≥ 14.  ... 
doi:10.1002/(sici)1097-0118(199709)26:1<53::aid-jgt6>3.0.co;2-g fatcat:xfa2d2v6jrbmxhnrn7t7vxjffy

Total Colourings - A survey [article]

Geetha Jayabalan and Narayanan N and K Somasundaram
2018 arXiv   pre-print
Vizing and Behzed conjectured that the total coloring can be done using at most Δ(G)+2 colors, where Δ(G) is the maximum degree of G. It is not settled even for planar graphs.  ...  In this paper we give a survey on total coloring of graphs.  ...  sufficiently large maximum degree.  ... 
arXiv:1812.05833v1 fatcat:2igatxvf55bcncqp2q2riwpp2y

Hard coloring problems in low degree planar bipartite graphs

Miroslav Chlebík, Janka Chlebíková
2006 Discrete Applied Mathematics  
In this paper we prove that the PRECOLORING EXTENSION problem on graphs of maximum degree 3 is polynomially solvable, but even its restricted version with 3 colors is NP-complete on planar bipartite graphs  ...  of maximum degree 4.  ...  Namely, in Section 3 we prove NP-completeness of PRECOLORING EXTENSION with 3 colors on planar bipartite graphs of maximum degree 4.  ... 
doi:10.1016/j.dam.2006.03.014 fatcat:isvodvt5njbf7jac5w3w5ubehy

Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers

Patcharapan Jumnongnit, Kittikorn Nakprasit
2017 International Journal of Mathematics and Mathematical Sciences  
In 2014, Dong and Wang obtained the results about tndi∑ (G) depending on the value of maximum average degree.  ...  Let G be a graph and ϕ:V(G)∪E(G)→{1,2,3,...,k} be a k-total coloring. Let w(v) denote the sum of color on a vertex v and colors assigned to edges incident to v.  ...  Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.  ... 
doi:10.1155/2017/5897049 fatcat:4fus6oy5xfhzxifqsllnv7hqei

Thoroughly Distributed Colorings [article]

Wayne Goddard, Michael A. Henning
2016 arXiv   pre-print
Equivalently, every color is a total dominating set. We define (G) as the maximum number of colors in such a coloring and (G) as the fractional version thereof.  ...  Further, although there are arbitrarily large examples of connected, cubic graphs with (G)=1, we show that for a connected cubic graph (G) > 2-o(1), and conjecture that it is always at least 2.  ...  We will show (Theorem 12) that there are graphs G with arbitrarily large minimum degree with FTD(G) < 1 + .  ... 
arXiv:1609.09684v1 fatcat:v7mi5bjirrhobpkxrnpkh3xhne

Coloring squares of planar graphs with girth six

Zdeněk Dvořák, Daniel Král', Pavel Nejedlý, Riste Škrekovski
2008 European journal of combinatorics (Print)  
We show that the conjecture for g = 6 is off by just one, i.e., the square of a planar graph G of girth at least six and sufficiently large maximum degree is (∆ + 2)-colorable.  ...  Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆ + 1)-colorable.  ...  The fourth author was supported in part by the Ministry of Higher Education, Science and Technology of Slovenia, Research Program P1-0297.  ... 
doi:10.1016/j.ejc.2007.11.005 fatcat:corodq372nh35fwzyknzwig3na

Graph coloring with no large monochromatic components

Nathan Linial, Jiří Matoušek, Or Sheffet, Gábor Tardos
2007 Electronic Notes in Discrete Mathematics  
We show that there are n-vertex 7-regular graphs G with mcc 2 (G) = Ω(n), and more sharply, for every ε > 0 there exists c ε > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ε for  ...  Haxell, Szabó, and Tardos proved mcc 2 (G) ≤ 20000 for every graph G of maximum degree 5.  ...  On the other hand, the HEX lemma shows that that for graphs G with maximum degree 6 mcc 2 (G) can be arbitrarily large.  ... 
doi:10.1016/j.endm.2007.07.020 fatcat:mhwm6nimc5gq3gxn56li4ggwlu

(d,1)-total labelling of planar graphs with large girth and high maximum degree

Fabrice Bazzaro, Mickaël Montassier, André Raspaud
2007 Discrete Mathematics  
In this paper, we prove that T d (G) + 2d − 2 for planar graphs with large girth and high maximum degree . Our results are optimal for d = 2.  ...  Havet, (d, 1)-total labelling of graphs, in: Workshop on Graphs and Algorithms, Dijon, France, 2003].  ...  By our results, we extend in a certain sense the results of Borodin et al. concerning the total coloring of planar graphs with large girth [4] and with large maximum degree [2] ; by the way, we will  ... 
doi:10.1016/j.disc.2005.12.059 fatcat:wnbb3w2qhnhsdjz54thdsykrb4

Algorithms [chapter]

2011 Graph Coloring Problems  
of Hamilton Cycles . 82 4.6 Brooks' Theorem for Triangle-Free Graphs 83 4.7 Graphs Without Large Complete Subgraphs 85 4.8 ^-Chromatic Graphs of Maximum Degree k 85 4.9 Total Coloring 86  ...  Graphs 97 5.1 Critical Graphs With Many Edges 97 5.2 Minimum Degree of 4-and 5-Critical Graphs > 98 5.3 Critical Graphs With Few Edges 99 5.4 Four-Critical Amenable Graphs 101 5.5 Four-Critical  ... 
doi:10.1002/9781118032497.ch10 fatcat:374tktuvgvekni4fnz3dgbytjm

Total colorings of planar graphs without adjacent triangles

Xiang-Yong Sun, Jian-Liang Wu, Yu-Wen Wu, Jian-Feng Hou
2009 Discrete Mathematics  
Let G be a planar graph without adjacent 3-cycles, that is, two cycles of length 3 are not incident with a common edge.  ...  In this paper, it is proved that the total coloring conjecture is true for G; moreover, if ∆(G) ≥ 9, then the total chromatic number χ (G) of G is ∆(G) + 1.  ...  Planar graphs with large maximum degree and without adjacent triangles Borodin, Kostochka and Woodall [1] proved that a planar graph G with maximum degree ∆ ≥ 11 has χ (G) = ∆(G) + 1, and they also obtained  ... 
doi:10.1016/j.disc.2007.12.071 fatcat:2ghs2m2ykbfxnlumiobfy6mgny

(2,1)-Total labeling of planar graphs with large maximum degree [article]

Yong Yu, Xin Zhang, Guanghui Wang, Jinbo Li
2011 arXiv   pre-print
In this paper, we prove that, for planar graph G with maximum degree Δ≥12 and d=2, the (2,1)-total labelling number λ_2^T(G) is at most Δ+2.  ...  The (d,1)-total labelling of graphs was introduced by Havet and Yu.  ...  In [1] , Bazzaro, Montassier and Raspaud proved a theorem for planar graph with large girth and high maximum degree: Theorem 1.2 ([1] ). Let G be a planar graph with maximum degree ∆ and girth g.  ... 
arXiv:1105.1908v1 fatcat:6ftmhmvlhfdavhk6nhosawsvqe

An annotated bibliography on 1-planarity

Stephen G. Kobourov, Giuseppe Liotta, Fabrizio Montecchiani
2017 Computer Science Review  
This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs.  ...  The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge.  ...  Every 1-planar graph with maximum vertex degree ∆ has a list total coloring with ∆ + 2 colors if ∆ ≥ 16 and with ∆ + 1 colors if ∆ ≥ 21 [139] .  ... 
doi:10.1016/j.cosrev.2017.06.002 fatcat:sjx2s5s4a5awzpsylfoc5upfmi

A structure of 1-planar graph and its applications to coloring problems [article]

Xin Zhang, Bei Niu, Jiguo Yu
2019 arXiv   pre-print
More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the (p,1)-total labelling number  ...  of every 1-planar graph G is at most Δ(G)+2p-2 provided that Δ(G)≥ 8p+2 and p≥ 2, and show that every 1-planar graph has an equitable edge coloring with k colors for any integer k≥ 18.  ...  For the edge and the total colorings of 1-planar graphs, Zhang and Wu [19] proved that the edge chromatic number of every 1-planar graph with maximum degree ∆ ≥ 10 is equal to ∆, and Zhang and Liu [  ... 
arXiv:1902.08945v1 fatcat:vfpzxlbkiferxdgtbixuvsn45e

Applying graph coloring in resour ce coordination for a high-density wireless environment

Li Zheng, Doan B. Hoang
2008 2008 8th IEEE International Conference on Computer and Information Technology  
The graph coloring solution algorithm for the OBSS group coordination assignment is presented. The actual coloring is demonstrated, using a heuristics of Maximum Degree First.  ...  In this paper we show that an OBSS environment can be modeled by a planar graph and the OBSS group coordination assignment problem can be considered as a vertex coloring problem whose solution involves  ...  A coloring example of a planar graph with 3 colors, using a heuristics of Maximum Degree First, has also been given.  ... 
doi:10.1109/cit.2008.4594754 dblp:conf/IEEEcit/ZhengH08 fatcat:ydnmsy275janxjesfa6ucuiuyu

Improved Induced Matchings in Sparse Graphs [chapter]

Rok Erman, Łukasz Kowalik, Matjaž Krnc, Tomasz Waleń
2009 Lecture Notes in Computer Science  
Induced matchings in sparse graphs Lower bounds, large minimum deg. Conclusion Let G be an n-vertex planar graph of minimum degree δ and let M be a maximum cardinality induced matching in G .  ...  Nishiezeki and Baybars, 1979 Let G be an n-vertex planar graph of minimum degree δ and let M be a maximum cardinality matching in G .  ... 
doi:10.1007/978-3-642-11269-0_11 fatcat:ytja66rr3ndq3dvbhkvuomi7ey
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