Clique-Relaxed Competitive Graph Coloring

In this paper we investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color α is legal for a vertex if said vertex has no neighbors colored α. Other variations of the game change this definition of a legal color. For a fixed color set, Alice wins the game if all vertices are colored when the game ends, while Bob wins if there is a point in the game in which a vertex cannot be assigned a legal color. 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 χg(G). A well studied variation is the d-relaxed coloring game [5] in which a legal coloring of a graph G is defined as any assignment of colors to V (G) such that the subgraph of G induced by any color class has maximum degree d. 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. In other words, a k-clique-relaxed n-coloring of G is an assignment of n colors to V (G) in which there are no monochromatic (k + 1)-cliques. For a fixed color set, the k-clique-relaxed coloring game begins with Alice coloring any vertex. Alice and Bob then alternate coloring vertices of G such that at no point in the game does there exists a monochromatic (k+1)-clique. Again, Alice wins the game if it terminates with all vertices colored, while Bob wins otherwise. The smallest number of colors needed for Alice to have a winning strategy on G is called the k-clique-relaxed game chromatic number of G, and is denoted χ (k) g (G). We use a modification of a well known strategy for the original coloring game in order to construct a winning strategy for Alice. Furthermore, we focus on the game as it is played on chordal graphs and partial k-trees. We extend results from [6] and [2] by specifically looking at the k-clique-relaxed coloring game as played on partial k-trees.