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Relaxed game chromatic number of graphs

Chun-Yen Chou, Weifan Wang, Xuding Zhu
2003 Discrete Mathematics  
This paper discusses a coloring 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.  ... 
doi:10.1016/s0012-365x(02)00521-6 fatcat:woapwpccivccbl43u37apm5uta

The 6-relaxed game chromatic number of outerplanar graphs

Jiaojiao Wu, Xuding Zhu
2008 Discrete Mathematics  
The d-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.  ... 
doi:10.1016/j.disc.2007.11.015 fatcat:xm5pe667jbdrrlvfemmeznrmji

Relaxed game chromatic number of trees and outerplanar graphs

Wenjie He, Jiaojiao Wu, Xuding Zhu
2004 Discrete Mathematics  
This paper studies the 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.  ... 
doi:10.1016/j.disc.2003.08.006 fatcat:howshc2mwjb75elp3nnle4cr2y

Complete Multipartite Graphs and the Relaxed Coloring Game

Charles Dunn
2011 Order  
The least k such that Alice has a winning strategy is called the d-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.  ... 
doi:10.1007/s11083-011-9217-1 fatcat:uwz7nlzz6nb5jof4ziftrg2m2a

Clique-relaxed graph coloring

Charles Dunn, Jennifer Firkins Nordstrom, Cassandra Naymie, Erin Pitney, William Sehorn, Charlie Suer
2011 Involve. A Journal of Mathematics  
We define a generalization 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.  ... 
doi:10.2140/involve.2011.4.127 fatcat:626zgudu5fg4pal2puriu5ixf4

Relaxed very asymmetric coloring games

Daqing Yang
2009 Discrete Mathematics  
The d-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.  ... 
doi:10.1016/j.disc.2007.11.058 fatcat:mp6pcco445eedo6aqjq24cfana

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.  ... 

On the Relaxed Colouring Game and the Unilateral Colouring Game

Elżbieta Sidorowicz
2013 Graphs and Combinatorics  
We prove that the P-unilateral 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)  ... 
doi:10.1007/s00373-013-1326-8 fatcat:fs52isaj7zc5tkwzl5qkccmpfy

Clique-Relaxed Competitive Graph Coloring [article]

Michel Alexis, Davis Shurbert, Charles Dunn, Jennifer Nordstrom
2014 arXiv   pre-print
The least 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.  ... 
arXiv:1412.3084v1 fatcat:qrld5okvqzh5plshofxxuuzuza

Weak acyclic coloring and asymmetric coloring games

H.A. Kierstead
2006 Discrete Mathematics  
The (a, b)-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.  ... 
doi:10.1016/j.disc.2004.07.042 fatcat:2gsjk6rlkrda5irmmjwk23tk7u

Directed defective asymmetric graph coloring games

Stephan Dominique Andres
2010 Discrete Applied Mathematics  
The smallest cardinality 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.  ... 
doi:10.1016/j.dam.2009.04.025 fatcat:gkyzayyy2rh5vajlw7jaj55ijm

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

Stephan Dominique Andres, Winfried Hochstättler, Christiane Schallück
2011 Discrete Applied Mathematics  
We prove that the 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.  ... 
doi:10.1016/j.dam.2010.05.003 fatcat:im47qa6ctjfgjlt3pjg5rkyzru

Adapted game colouring of graphs

H.A. Kierstead, Chung-Ying Yang, Daqing Yang, Xuding Zhu
2012 European journal of combinatorics (Print)  
We prove that the maximum adapted 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.  ... 
doi:10.1016/j.ejc.2011.11.001 fatcat:havlvtufujezdlmdvwi533fvre

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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