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Total colorings of planar graphs with large maximum degree

1997
*
Journal of Graph Theory
*

It is proved that a

doi:10.1002/(sici)1097-0118(199709)26:1<53::aid-jgt6>3.0.co;2-g
fatcat:xfa2d2v6jrbmxhnrn7t7vxjffy
*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. ...##
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Total Colourings - A survey
[article]

2018
*
arXiv
*
pre-print

Vizing and Behzed conjectured that the

arXiv:1812.05833v1
fatcat:2igatxvf55bcncqp2q2riwpp2y
*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*. ...##
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Hard coloring problems in low degree planar bipartite graphs

2006
*
Discrete Applied Mathematics
*

In this paper we prove that the PRECOLORING EXTENSION problem on

doi:10.1016/j.dam.2006.03.014
fatcat:isvodvt5njbf7jac5w3w5ubehy
*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. ...##
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Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers

2017
*
International Journal of Mathematics and Mathematical Sciences
*

In 2014, Dong and Wang obtained the results about tndi∑ (G) depending on the value

doi:10.1155/2017/5897049
fatcat:4fus6oy5xfhzxifqsllnv7hqei
*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. ...##
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Thoroughly Distributed Colorings
[article]

2016
*
arXiv
*
pre-print

Equivalently, every

arXiv:1609.09684v1
fatcat:v7mi5bjirrhobpkxrnpkh3xhne
*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 + . ...##
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Coloring squares of planar graphs with girth six

2008
*
European journal of combinatorics (Print)
*

We show that the conjecture for g = 6 is off by just one, i.e., the square

doi:10.1016/j.ejc.2007.11.005
fatcat:corodq372nh35fwzyknzwig3na
*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. ...##
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Graph coloring with no large monochromatic components

2007
*
Electronic Notes in Discrete Mathematics
*

We show that there are n-vertex 7-regular

doi:10.1016/j.endm.2007.07.020
fatcat:mhwm6nimc5gq3gxn56li4ggwlu
*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*. ...##
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(d,1)-total labelling of planar graphs with large girth and high maximum degree

2007
*
Discrete Mathematics
*

In this paper, we prove that T d (G) + 2d − 2 for

doi:10.1016/j.disc.2005.12.059
fatcat:wnbb3w2qhnhsdjz54thdsykrb4
*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 ...##
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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 ...

##
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Total colorings of planar graphs without adjacent triangles

2009
*
Discrete Mathematics
*

Let G be a

doi:10.1016/j.disc.2007.12.071
fatcat:2ghs2m2ykbfxnlumiobfy6mgny
*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 ...##
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(2,1)-Total labeling of planar graphs with large maximum degree
[article]

2011
*
arXiv
*
pre-print

In this paper, we prove that, for

arXiv:1105.1908v1
fatcat:6ftmhmvlhfdavhk6nhosawsvqe
*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. ...##
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An annotated bibliography on 1-planarity

2017
*
Computer Science Review
*

This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview

doi:10.1016/j.cosrev.2017.06.002
fatcat:sjx2s5s4a5awzpsylfoc5upfmi
*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] . ...##
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A structure of 1-planar graph and its applications to coloring problems
[article]

2019
*
arXiv
*
pre-print

More precisely, we verify the well-known List Edge

arXiv:1902.08945v1
fatcat:vfpzxlbkiferxdgtbixuvsn45e
*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 [ ...##
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Applying graph coloring in resour ce coordination for a high-density wireless environment

2008
*
2008 8th IEEE International Conference on Computer and Information Technology
*

The

doi:10.1109/cit.2008.4594754
dblp:conf/IEEEcit/ZhengH08
fatcat:ydnmsy275janxjesfa6ucuiuyu
*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. ...##
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Improved Induced Matchings in Sparse Graphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

Induced matchings in sparse

doi:10.1007/978-3-642-11269-0_11
fatcat:ytja66rr3ndq3dvbhkvuomi7ey
*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 . ...
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