We define a generalization of the chromatic number of a graph called the -clique-relaxed chromatic number, denoted . We prove bounds on for all graphs , including corollaries for outerplanar and planar graphs. We also define the -clique-relaxed game chromatic number, , of a graph . We prove for all outerplanar graphs , and give an example of an outerplanar graph with . Finally, we prove that if is a member of a particular subclass of outerplanar graphs, then .
"Clique-relaxed graph coloring." Involve 4 (2) 127 - 138, 2011. https://doi.org/10.2140/involve.2011.4.127