Asymmetric coloring games on incomparability graphs

TL;DR

This paper investigates asymmetric coloring games on incomparability graphs of posets, introducing new parameters and conjecturing their boundedness based on poset width, while providing partial proofs and counterexamples.

Contribution

It introduces the $(a,b)$-game chromatic and Grundy numbers for incomparability graphs and conjectures their boundedness in relation to poset width, offering necessary conditions and evidence.

Findings

Necessary condition for boundedness of parameters

Counterexample showing unboundedness of game chromatic number in terms of Grundy number

Evidence supporting the conjecture relating parameters to poset width

Abstract

Consider the following game on a graph $G$: Alice and Bob take turns coloring the vertices of $G$ properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of $G$ is the minimum number of colors that allows Alice to win the game. The game Grundy number of $G$ is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The $(a,b)$-game chromatic and Grundy numbers are defined likewise except that Alice colors $a$ vertices and Bob colors $b$ vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs $(a,b)$ for which the $(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu.