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The Game Coloring Number of Planar Graphs

1999
*
Journal of combinatorial theory. Series B (Print)
*

We show that

doi:10.1006/jctb.1998.1878
fatcat:opm6xhs3qrgmthlldvdtqapcme
*the**game**coloring**number**of*a*planar**graph*is at most 19. ... This paper discusses a variation*of**the**game*chromatic*number**of*a*graph*:*the**game**coloring**number*. This parameter provides an upper bound for*the**game*chromatic*number**of*a*graph*. ... Since*the*acyclic chromatic*number**of*a*planar**graph*is at most 5, it follows that*the**game*chromatic*number**of*a*planar**graph*is at most 30. ...##
###
Bounds for the game coloring number of planar graphs with a specific girth
[article]

2016
*
arXiv
*
pre-print

Let col_g(G) be

arXiv:1610.01260v2
fatcat:slimz42zfjexrn7u4xnajtub6i
*the**game**coloring**number**of*a given*graph*G. Define*the**game**coloring**number**of*a family*of**graphs*H as col_g(H) := { col_g(G):G ∈H}. ... Let P_k be*the*family*of**planar**graphs**of*girth at least k. We show that col_g(P_7) ≤ 5. This result extends a result about*the**coloring**number*by Wang and Zhang WZ11 ( col_g(P_8) ≤ 5). ...*The*second author was supported by*the*Commission on Higher Education and*the*Thailand Research Fund under grant RSA5780014. ...##
###
The Hats game. The power of constructors
[article]

2021
*
arXiv
*
pre-print

We present an example

arXiv:2102.07138v1
fatcat:dyxf2f5hhjctbnrb6fnjzrdvby
*of*a*planar**graph*for which*the*sages win for k = 14. We also give an easy proof*of**the*theorem about*the*Hats*game*on "windmill"*graphs*. ... We analyze*the*following general version*of**the*deterministic Hats*game*. Several sages wearing*colored*hats occupy*the*vertices*of*a*graph*. Each sage can have a hat*of*one*of*k*colors*. ...*The*relation between*the**number*HG(G) and*the**planarity**of**graph*G is one*of**the*open problems. Namely, does there exist a*planar**graph*with an arbitrary large hat guessing*number*? ...##
###
The Map-Coloring Game

2007
*
The American mathematical monthly
*

We would like to thank Steven Brams for useful materials and comments on

doi:10.1080/00029890.2007.11920471
fatcat:6p5buz5zgjgcrgu2zrjerivanq
*the*origin*of**the*map-*coloring**game*. ...*The*research*of**the*fourth author is supported in part by*the*National Science Council under grant NSC95-2115-M-110-013-MY3. ... 2-*coloring**numbers*. proved*the*Burr-Erdős conjecture for*planar**graphs*by showing that*the*2-*coloring**number**of*any*planar**graph*is at most 761. ...##
###
Planar graph coloring with an uncooperative partner

1994
*
Journal of Graph Theory
*

We show that

doi:10.1002/jgt.3190180605
fatcat:j37ekqgi5ndp3hrqra3gpg44iy
*the**game*chromatic*number**of*a*planar**graph*is at most 33. ... In particular,*the**game*chromatic*number**of*a*graph*is bounded in terms*of*its genus. ... ACKNOWLEDGMENT*The*authors would like to express their appreciation to Nate Dean, Paul Seymour, Neil Robertson, and Robin Thomas for stimulating conversations on*the*interplay between*game*chromatic*number*...##
###
A bound for the game chromatic number of graphs

1999
*
Discrete Mathematics
*

*the*

*game*chromatic

*number*

*of*

*planar*

*graphs*. ... In particular, since a

*planar*

*graph*has acyclic chromatic

*number*at most 5, we conclude that

*the*

*game*chromatic

*number*

*of*a

*planar*

*graph*is at most 30, which improves

*the*previous known upper bound for ... This implies, in particular, that

*the*

*game*chromatic

*number*

*of*a

*planar*

*graph*is at most 30. ...

##
###
Planar graphs decomposable into a forest and a matching

2009
*
Discrete Mathematics
*

He, Hou, Lih, Shao, Wang, and Zhu showed that a

doi:10.1016/j.disc.2007.12.104
fatcat:tc2hrgmzszehtnjddo7g36ukde
*planar**graph**of*girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved*the*bound on girth to 9. ... We give sufficient conditions for a*planar**graph*with 3-cycles to be decomposable into a forest and a matching. ...*The*third author's research was supported in part by*the*grant DMS-0650784. ...##
###
The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses
[article]

2015
*
arXiv
*
pre-print

We denote by χ g (G)

arXiv:1507.03195v1
fatcat:34nn5523jjeafj2bzp33spsrtm
*the**game*chromatic*number**of*a*graph*G, which is*the*smallest*number**of**colors*Alice needs to win*the**coloring**game*on G. We know from Montassier et al. [M. Montassier, P. ... Decomposing a*planar**graph*with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that*planar**graphs*with girth at least 8 have*game*chromatic*number*at most 5. ... that every (1, k)-decomposable*graph*has χ g (G) ≤ k + 4, then deduced upper bounds for*the**game*chromatic*number**of**planar**graphs*with given girth. ...##
###
Edge-disjoint odd cycles in graphs with small chromatic number

1999
*
Annales de l'Institut Fourier
*

For 4-

doi:10.5802/aif.1691
fatcat:cfl5s6neqjdnnm4hf4y7ceudk4
*colorable**graphs*, and in particular for*planar**graphs*, no simple structural property has been found so far to see that*the**game*is unfair.*The*Konig property. ... At*the*end*of**the**game*, all*the*edges*of*G have been*colored*;*the*red edges define a partial*graph*GR with no odd cycles and*the*blue edges define a partial*graph*GB with no odd cycles. ...##
###
On-line Ramsey Theory

2004
*
Electronic Journal of Combinatorics
*

*The*question

*of*whether

*planar*

*graphs*are self-unavoidable is left open. We also consider a multicolor version

*of*Ramsey on-line

*game*. ... In particular, we prove that Builder has a winning strategy for any $k$-

*colorable*

*graph*$H$ in

*the*

*game*played on $k$-

*colorable*

*graphs*. ... Let n be

*the*

*number*

*of*vertices

*of*a 3-

*colorable*

*graph*F on which Builder wins against Painter armed in c − 1

*colors*. ...

##
###
Page 3162 of Mathematical Reviews Vol. , Issue 95f
[page]

1995
*
Mathematical Reviews
*

*The*question whether

*planar*

*graphs*have bounded

*game*chro- matic

*number*is resolved in this paper: it is shown that

*the*

*game*chromatic

*number*

*of*a

*planar*

*graph*is at most 33. ... In particular,

*the*

*game*chro- matic

*number*

*of*a

*game*is bounded in terms

*of*its genus. ...

##
###
The game coloring number of pseudo partial k-trees

2000
*
Discrete Mathematics
*

We discuss

doi:10.1016/s0012-365x(99)00237-x
fatcat:a55yzwogxfgefmhcbsttpre6ie
*the**game**coloring**number*(as well as*the**game*chromatic*number*)*of*(a; b)-pseudo partial k-trees, and prove that*the**game**coloring**number**of*an (a; b)-pseudo partial k-tree is at most 3k + 2a ... , outerplanar*graphs*and*planar**graphs*. ... It follows from Corollary 2 that*the**game**coloring**number**of*an outer-*planar**graph*is at most 8. It was proved in [5] that*the**game**coloring**number**of*an outer-*planar**graph*is at most 7. ...##
###
The game of arboricity

2005
*
Discrete Mathematics & Theoretical Computer Science
*

International audience Using a fixed set

doi:10.46298/dmtcs.3428
fatcat:a3thpfvn35hjnb2zc6gpfohgqu
*of**colors*$C$, Ann and Ben*color**the*edges*of*a*graph*$G$ so that no monochromatic cycle may appear. ...*The*upper bound is achieved by a suitable version*of**the*activation strategy, used earlier for*the*vertex*coloring**game*. We also provide other strategie based on induction. ...*The**game**of*arboricity Introduction We consider*the*following*graph**coloring**game*. Ann and Ben alternately*color**the*edges*of*a*graph*G using a fixed set*of**colors*C. ...##
###
The coloring game on planar graphs with large girth, by a result on sparse cactuses

2017
*
Discrete Mathematics
*

We denote by χ g (G)

doi:10.1016/j.disc.2016.08.010
fatcat:gcddz4mwx5eifcjemqmf2roeni
*the**game*chromatic*number**of*a*graph*G, which is*the*smallest*number**of**colors*Alice needs to win*the**coloring**game*on G. We know from Montassier et al. [M. Montassier, P. ... Decomposing a*planar**graph*with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that*planar**graphs*with girth at least 8 have*game*chromatic*number*at most 5. ... This ends*the*proof*of*Theorem 1. ...##
###
Page 8410 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

This is

*the*first upper bound for*the**game**coloring**number**of*such*graphs*, and it also improves considerably*the*previous known upper bound for*the**game*chro- matic*number**of*such*graphs*. ... We discuss*the**game**coloring**number*(as well as*the**game*chromatic*number*)*of*(a,b)-pseudo partial k-trees, and prove that*the**game**color*- ing*number**of*an (a, b)-pseudo partial k-tree is at most 3k + ...
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