We define a generalization of the chromatic number of a graph [math] called the [math] -clique-relaxed chromatic number, denoted [math] . We prove bounds on [math] for all graphs [math] , including corollaries for outerplanar and planar graphs. We also define the [math] -clique-relaxed game chromatic number, [math] , of a graph [math] . We prove [math] for all outerplanar graphs [math] , and give an example of an outerplanar graph [math] with [math] . Finally, we prove that if [math] is a member of a particular subclass of outerplanar graphs, then [math] .