Adapted game colouring of graphs

  • Authors:

  • H. A. Kierstead;Chung-Ying Yang;Daqing Yang;Xuding Zhu

  • Affiliations:

  • Department of Mathematics and Statistics, Arizona State University, Tempe AZ 85287, United States;Department of Applied Mathematics, National Sun Yat-sen University, Taiwan;Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian 350002, China;Department of Mathematics, Zhejiang Normal University, Jinhua, China

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2012

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Abstract

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 @g"a"d"g(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.