Relaxed game chromatic number of graphs

  • Authors:

  • Chun-Yen Chou;Weifan Wang;Xuding Zhu

  • Affiliations:

  • Department of Mathematics Education, National Hualien Teachers College, 123 Hua-Hsi, Hualien, Taiwan;Department of Mathematics, Liaoning University, Shenyang 110036, China;Department of Applied Mathematics, National Sun Yat-sen University, Taiwan

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2003

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Abstract

This paper discusses a coloring game on graphs. Let k, d be non-negative integers and C a set of k colors. Two persons, Alice and Bob, alternately color the vertices of G with colors from C, with Alice having the first move. A color i is legal for an uncolored vertex x if by coloring x with color i, the subgraph of G induced by those vertices of color i has maximum degree at most d. 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. Alice's goal is to produce a legal coloring which colors all the vertices of G, and Bob's goal is to prevent this from happening. We shall prove that if G is a forest, then for k = 3, d ≥ 1, Alice has a winning strategy. If G is an outerplanar graph, then for k = 6 and d ≥ 1, Alice has a winning strategy.