Online coloring a token graph
Abstract
We study a combinatorial coloring game between two players, Spoiler and Algorithm, who alternate turns. First, Spoiler places a new token at a vertex in $G$, and Algorithm responds by assigning a color to the new token. Algorithm must ensure that tokens on the same or adjacent vertices receive distinct colors. Spoiler must ensure that the token graph (in which two tokens are adjacent if and only if their distance in $G$ is at most $1$) has chromatic number at most $w$. Algorithm wants to minimize the number of colors used, and Spoiler wants to force as many colors as possible. Let $f(w,G)$ be the minimum number of colors needed in an optimal Algorithm strategy. A graph $G$ is onlineperfect if $f(w,G) = w$. We give a forbidden induced subgraph characterization of the class of onlineperfect graphs. When $G$ is not onlineperfect, determining $f(w,G)$ seems challenging; we establish $f(w,G)$ asymptotically for some (but not all) of the minimal graphs that are not onlineperfect. The game is motivated by a natural online coloring problem on the real line which remains open.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.08699
 arXiv:
 arXiv:1712.08699
 Bibcode:
 2017arXiv171208699M
 Keywords:

 Mathematics  Combinatorics;
 05C57 (Primary) 05C15 (Secondary)
 EPrint:
 11 pages, 1 figure