Online coloring a token graph

TL;DR

This paper investigates an online graph coloring game involving token placement and coloring, characterizes online-perfect graphs, and explores the complexity of the minimum colors needed in non-perfect cases.

Contribution

It provides a forbidden induced subgraph characterization of online-perfect graphs and analyzes the asymptotic behavior of the coloring function for non-perfect graphs.

Findings

Characterization of online-perfect graphs via forbidden subgraphs

Asymptotic analysis of the minimum colors for certain non-perfect graphs

Open problem related to online coloring on the real line

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 online-perfect if $f(w,G) = w$. We give a forbidden induced subgraph characterization of the class of online-perfect graphs. When $G$ is not online-perfect, determining $f(w,G)$ seems challenging; we establish $f(w,G)$ asymptotically for some (but not all) of the minimal graphs that are not online-perfect. The game is motivated by a natural online coloring problem on the real line which remains open.