Discrete Mathematics and Combinatorics
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, χg(k)(G), of a graph G. We prove χg(2)(G)≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χg(2)(H) ≥ 3. Finally, we prove that if H is a member of a particular subclass of outerplanar graphs, then χg(2)(H) ≤ 3.
Copyright ©2011 by Mathematical Sciences Publishers
Charles Dunn, Jennifer Firkins Nordstrom, Cassandra Naymie, Erin Pitney, William Sehorn, & Charlie Suer
Clique-relaxed graph coloring
Involve, 2011, volume 4, issue 2, pages 127-138
Dunn, Charles; Nordstrom, Jennifer Firkins; Naymie, Cassandra; Pitney, Erin; Sehorn, William; and Suer, Charlie, "Clique-Relaxed Graph Coloring" (2011). Faculty Publications. Published Version. Submission 4.