Clique-Relaxed Competitive Graph Coloring

TL;DR

This paper explores a variation of the graph coloring game where the goal is to color vertices so that no monochromatic (k+1)-cliques form, extending previous relaxations by considering maximum clique size constraints.

Contribution

It introduces the k-clique-relaxed n-coloring game, a novel variation that limits monochromatic clique sizes, generalizing existing relaxed coloring concepts.

Findings

Defines the k-clique-relaxed n-coloring game and its rules.

Establishes bounds and strategies for the game.

Connects the new game to existing coloring and game chromatic number concepts.

Abstract

We investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color {\alpha} is legal for a vertex if said vertex has no neighbors colored {\alpha}. Other variations of the game change this definition of a legal color. For a fixed color set, Alice wins the game if all vertices are colored when the game ends, while Bob wins if there is a point in the game in which a vertex cannot be assigned a legal color. The least number of colors needed for Alice to have a winning strategy on a graph G is called the game chromatic number of G, and is denoted \c{hi}g(G). A well studied variation is the d-relaxed coloring game [5] in which a legal coloring of a graph G is defined as any assignment of colors to V (G) such that the subgraph of G induced by any color class has maximum degree d. We focus on the k-clique-relaxed n-coloring game. A k-clique-relaxed n-coloring of a graph G is an n-coloring in which the subgraph of G induced by any color class has maximum clique size k or less. In other words, a k-clique-relaxed n-coloring of G is an assignment of n colors to V (G) in which there are no monochromatic (k + 1)-cliques.