Autotopism stabilized colouring games on rook's graphs

Based on the fact that every partial colouring of the rook’s graph Kr✷Ks is uniquely related to an r × s partial Latin rectangle, this work deals with the Θ-stabilized colouring game on the graph Kr✷Ks. This is a variant of the classical colouring game on finite graphs [1,2,6,7] so that each move must respect a given autotopism Θ of the resulting partial Latin rectangle. The complexity of this variant is examined by means of its Θ-stabilized game chromatic number, which depends in turn on the cycle structure of the autotopism under consideration. Based on the known classification of such cycle structures [3,4,5,8], we determine in a constructive way the game chromatic number associated to those rook’s graphs Kr✷Ks, for which r ≤ s ≤ 8