Abstract

We define a generalization of the chromatic number of a graph G called the k-clique-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_g^(k)(G), of a graph G. We prove χ_g⁽²⁾(G) ≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χ_g⁽²⁾(H) ≥ 3. Finally, we prove that if H is a member of a particular subclass of outerplanar graphs, then χ_g⁽²⁾(H) ≤ 3.

Keywords

Mathematical Subject Classification 2000

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