TY - JOUR

T1 - Adapted game colouring of graphs

AU - Kierstead, Henry

AU - Yang, Chung Ying

AU - Yang, Daqing

AU - Zhu, Xuding

N1 - Funding Information:
H. A. Kierstead supported in part by NSA grant MDA 904-03-1-0007 and NSF grant DMS-0901520 . Daqing Yang supported in part by NSFC under grants 10771035 and 10931003 , grant JA10018 of Fujian. Xuding Zhu supported in part by grants NSFC No. 11171730 and ZJNSF No. Z6110786 .

PY - 2012/5

Y1 - 2012/5

N2 - Suppose G=(V, E) is a graph and F is a colouring of its edges (not necessarily proper) that uses the colour set X. In an adapted colouring game, Alice and Bob alternately colour uncoloured vertices of G with colours from X. A partial colouring c of the vertices of G is legal if there is no edge e=xy such that c(x)=c(y)=F(e). In the process of the game, each partial colouring must be legal. If eventually all the vertices of G are coloured, then Alice wins the game. Otherwise, Bob wins the game. The adapted game chromatic number of a graph G, denoted by χ adg(G), is the minimum number of colours needed to ensure that for any edge colouring F of G, Alice has a winning strategy for the game. This paper studies the adapted game chromatic number of various classes of graphs. 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 2k+1, and the adapted game chromatic number of planar graphs is at most 11.

AB - Suppose G=(V, E) is a graph and F is a colouring of its edges (not necessarily proper) that uses the colour set X. In an adapted colouring game, Alice and Bob alternately colour uncoloured vertices of G with colours from X. A partial colouring c of the vertices of G is legal if there is no edge e=xy such that c(x)=c(y)=F(e). In the process of the game, each partial colouring must be legal. If eventually all the vertices of G are coloured, then Alice wins the game. Otherwise, Bob wins the game. The adapted game chromatic number of a graph G, denoted by χ adg(G), is the minimum number of colours needed to ensure that for any edge colouring F of G, Alice has a winning strategy for the game. This paper studies the adapted game chromatic number of various classes of graphs. 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 2k+1, and the adapted game chromatic number of planar graphs is at most 11.

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U2 - 10.1016/j.ejc.2011.11.001

DO - 10.1016/j.ejc.2011.11.001

M3 - Article

AN - SCOPUS:84855177730

VL - 33

SP - 435

EP - 445

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 4

ER -