Autotopism stabilized colouring games on rook’s graphs

Abstract

This paper deals with the Θ-stabilized colouring game on the n×n rook’s graph, which constitutes a variant of the classical colouring game on finite graphs so that each configuration of the game is uniquely related to a partial Latin square of order n that respects a given autotopism Θ. The complexity of this variant is examined by means of its Θ-stabilized game chromatic number, whose currently known upper bound is improved to 2n−1. Particularly, we introduce the concept of a passing board that enables us to describe a constructive method to compute this number. Based on this method, we explicitly determine the game chromatic number associated to those n×n rook’s graphs, for which n≤4.