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Relaxed game chromatic number of graphs

2003
*
Discrete Mathematics
*

This paper discusses a coloring

doi:10.1016/s0012-365x(02)00521-6
fatcat:woapwpccivccbl43u37apm5uta
*game*on*graphs*. Let k; d be non-negative integers and C a set*of*k colors. ... Each move*of*Alice or Bob colors an uncolored vertex with a legal color. The*game*is over if either all vertices are colored, or no more vertices can be colored with a legal color. ... Similarly, the 0-*relaxed*coloring*game*is the same as the coloring*game*, and the 0-*relaxed**game**chromatic**number**of*a*graph*G is the same as its*game**chromatic**number*. ...##
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The 6-relaxed game chromatic number of outerplanar graphs

2008
*
Discrete Mathematics
*

The d-

doi:10.1016/j.disc.2007.11.015
fatcat:xm5pe667jbdrrlvfemmeznrmji
*relaxed**game**chromatic**number**of*G, denoted by χ , is the least*number*k so that when playing the (k, d)-*relaxed*colouring*game*on G, Alice has a winning strategy. ... The (k, d)-*relaxed*colouring*game*on G is a*game*played by two players, Alice and Bob, who take turns colouring the vertices*of*G with legal colours from a set X*of*k colours. ... A 0-*relaxed*colouring*game*is the same as the colouring*game*, and the 0-*relaxed**game**chromatic**number**of*a*graph*G is called the*game**chromatic**number**of*G. ...##
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Relaxed game chromatic number of trees and outerplanar graphs

2004
*
Discrete Mathematics
*

This paper studies the

doi:10.1016/j.disc.2003.08.006
fatcat:howshc2mwjb75elp3nnle4cr2y
*relaxed**game**chromatic**number**of*trees and outerplanar*graphs*. It is proved that if G is a tree then d g (G) 6 2 for d ¿ 2. ... If G is an outerplanar*graph*, then d g (G) 6 5 for d ¿ 2, and d g (G) ¿ 3 for d 6 4. ... Introduction The*relaxed**game**chromatic**number**of*a*graph*is deÿned through a two-person*game*. Let G = (V; E) be a*graph*. Let k and d be positive integers. ...##
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Complete Multipartite Graphs and the Relaxed Coloring Game

2011
*
Order
*

The least k such that Alice has a winning strategy is called the d-

doi:10.1007/s11083-011-9217-1
fatcat:uwz7nlzz6nb5jof4ziftrg2m2a
*relaxed**game**chromatic**number**of*G, denoted χ d g (G). ... Let k be a positive integer, d be a nonnegative integer, and G be a finite*graph*. Two players, Alice and Bob, play a*game*on G by coloring the uncolored vertices with colors from a set X*of*k colors. ... The*relaxed**game**chromatic**number*has been considered in [3, 5, 6, 7, 9] . Suppose that Alice and Bob are playing the (k, d)-*relaxed*coloring*game*on a*graph*G. ...##
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Clique-relaxed graph coloring

2011
*
Involve. A Journal of Mathematics
*

We define a generalization

doi:10.2140/involve.2011.4.127
fatcat:626zgudu5fg4pal2puriu5ixf4
*of*the*chromatic**number**of*a*graph*G called the kclique-*relaxed**chromatic**number*, denoted χ (k) (G). ... We prove bounds on χ (k) (G) for all*graphs*G, including corollaries for outerplanar and planar*graphs*. We also define the k-clique-*relaxed**game**chromatic**number*, χ (k) g (G),*of*a*graph*G. ... The following theorem gives an upper bound for the k-clique-*relaxed**chromatic**number**of*a*graph*G in terms*of*the standard*chromatic**number**of*G. Theorem 1. Let G be a*graph*. ...##
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Relaxed very asymmetric coloring games

2009
*
Discrete Mathematics
*

The d-

doi:10.1016/j.disc.2007.11.058
fatcat:mp6pcco445eedo6aqjq24cfana
*relaxed*(a, b)-*game**chromatic**number*, denoted by (a, b)-χ d g (G),*of*G is the least r for which Alice has a winning strategy in the (r, d)-*relaxed*(a, b)-coloring*game*. ... This paper investigates a competitive version*of*the coloring*game*on a finite*graph*G. An asymmetric variant*of*the (r, d)*relaxed*coloring*game*is called the (r, d)-*relaxed*(a, b)-coloring*game*. ... The d-*relaxed**chromatic**number*, χ d (G),*of*a*graph*G is the least r such that the*graph*admits a d-*relaxed*proper r -coloring. ...##
###
Page 9575 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

The d-

*relaxed**game**chromatic**number*1,°(G) is determined by exactly the same*game*except that each vertex is allowed to have its color the same as up tc d*of*its neighbors so that 7,”(G) = Z(G). ... The principle result*of*the paper is that for each d > 8 and each outerplanar*graph*G the d-*relaxed**game**chromatic**number*satisfies xs (G) <2. This strongly answers the question raised by C.-Y. ...##
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On the Relaxed Colouring Game and the Unilateral Colouring Game

2013
*
Graphs and Combinatorics
*

We prove that the P-unilateral

doi:10.1007/s00373-013-1326-8
fatcat:fs52isaj7zc5tkwzl5qkccmpfy
*game**chromatic**number*is monotone and is the upper bound for the*game**chromatic**number*and the*relaxed**game**chromatic**number*. ... We find the upper bound for the S 2 -unilateral*game**chromatic**number*for*graphs*from H 3 and we study the S d -unilateral*game**chromatic**number*for*graphs*from H 4 for d ∈ {2, 3}. ... Open Access This article is distributed under the terms*of*the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) ...##
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Clique-Relaxed Competitive Graph Coloring
[article]

2014
*
arXiv
*
pre-print

The least

arXiv:1412.3084v1
fatcat:qrld5okvqzh5plshofxxuuzuza
*number**of*colors needed for Alice to have a winning strategy on a*graph*G is called the*game**chromatic**number**of*G, and is denoted ḩi̧g(G). ... We focus on the k-clique-*relaxed*n-coloring*game*. A k-clique-*relaxed*n-coloring*of*a*graph*G is an n-coloring in which the subgraph*of*G induced by any color class has maximum clique size k or less. ... the 2-clique-*relaxed**game**chromatic**number**of*any chordal*graph*with clique*number*3 is at most 3. ...##
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Weak acyclic coloring and asymmetric coloring games

2006
*
Discrete Mathematics
*

The (a, b)-

doi:10.1016/j.disc.2004.07.042
fatcat:2gsjk6rlkrda5irmmjwk23tk7u
*game**chromatic**number**of*G, denoted (a, b)g (G) , is the least integer t such that Alice has a winning strategy when the*game*is played on G using t colors. ... We introduce the notion*of*weak acyclic coloring*of*a*graph*. This is a*relaxation**of*the usual notion*of*acyclic coloring which is often sufficient for applications. ... Introduction The main purpose*of*this article is to bound the*game**chromatic**number**of**graphs*in terms*of*their acyclic*chromatic**number*. ...##
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Directed defective asymmetric graph coloring games

2010
*
Discrete Applied Mathematics
*

The smallest cardinality

doi:10.1016/j.dam.2009.04.025
fatcat:gkyzayyy2rh5vajlw7jaj55ijm
*of*a color set C with which player I has a winning strategy for the*game*is called d-*relaxed*(a, b)-*game**chromatic**number**of*D. ... ., b d+1 + 3) for the d-*relaxed*(a, b)-*game**chromatic**number**of*orientations*of*forests (resp., undirected forests) for any d and a ≥ b ≥ 1. ... Acknowledgements The comments*of*the anonymous referees and fruitful discussions with Winfried Hochstättler helped to improve the presentation*of*the paper. ...##
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Page 3162 of Mathematical Reviews Vol. , Issue 95f
[page]

1995
*
Mathematical Reviews
*

It is also shown that the

*game**chromatic**number**of*an outerplanar*graph*is at most 8, and lower bounds for the*game**chromatic*suniber*of*these classes*of**graphs*are given. ... 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. ...##
###
The game chromatic index of wheels

2011
*
Discrete Applied Mathematics
*

We prove that the

doi:10.1016/j.dam.2010.05.003
fatcat:im47qa6ctjfgjlt3pjg5rkyzru
*game**chromatic*index*of*n-wheels is n for n ≥ 6. ... This leaves a gap*of*6 for the maximum*game*colouring*number*and a gap*of*9 for the maximum*game**chromatic**number**of*planar*graphs*. ... We will focus here on the results concerning the*game**chromatic**number**of*line*graphs*, i.e. the results on the*game**chromatic*index*of**graphs*. ...##
###
Adapted game colouring of graphs

2012
*
European journal of combinatorics (Print)
*

We prove that the maximum adapted

doi:10.1016/j.ejc.2011.11.001
fatcat:havlvtufujezdlmdvwi533fvre
*game**chromatic**number**of*trees is 3, the maximum adapted*game**chromatic**number**of*outerplanar*graphs*is 5, the adapted*game**chromatic**number**of*partial k-trees is at most ... This paper studies the adapted*game**chromatic**number**of*various classes*of**graphs*. ... These colouring*games*define, respectively, the acyclic*game**chromatic**number*and the d-*relaxed**game**chromatic**number**of**graphs*. We can also consider list versions*of*these*games*. ...##
###
Page 7547 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

*chromatic*

*number*lies in the famous gap between the clique

*number*and the

*chromatic*

*number*), includ- ing a discussion

*of*list coloring and

*of*the fractional

*chromatic*

*number*

*of*the plane (it is proved ... For example, the problem

*of*determining whether the fractional

*chromatic*

*number*

*of*a

*graph*is at most r (for any fixed r > 2) is NP-complete. ...

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