Asymmetric Coloring Games on Incomparability Graphs

Abstract

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.

Hardness of some variants of the graph coloring game

“…Problem 6 of [9]: Is it true that Γ g (G) ≤ χ g (G) for every graph G? In 2015, Krawczyk and Walczak [12] answered Problem 5 of [9] in the negative: χ g (G) is not upper bounded by a function of Γ g (G). To the best of our knowledge, Problem 6 of [9] is still open.…”

Adversarial and Online Algorithms

“…Unfortunately infinitely many chains would be required for Alice to always win the chain decomposition game on every partial order. The following theorem was first proved by Krawczyk and Walczak 2015 [46] in the context of colouring incompatibility graphs (defined later). Theorem 4.6 (Krawczyk and Walczak 2015 [46]).…”

“…The following theorem was first proved by Krawczyk and Walczak 2015 [46] in the context of colouring incompatibility graphs (defined later). Theorem 4.6 (Krawczyk and Walczak 2015 [46]). For all 𝑘 there exists a partial order 𝑃 of width 2 such that 𝑘 ≤ 𝜔 𝑔 (𝑃).…”