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Colouring games based on autotopisms of Latin hyper-rectangles. (English) Zbl 1420.05114
Summary: Every partial colouring of a Hamming graph is uniquely related to a partial Latin hyper-rectangle. In this paper we introduce the \(\Theta\)-stabilized \((a,b)\)-colouring game for Hamming graphs, a variant of the \((a,b)\)-colouring game so that each move must respect a given autotopism \(\Theta\) of the resulting partial Latin hyperrectangle. We examine the complexity of this variant by means of its chromatic number. We focus in particular on the bi-dimensional case, for which the game is played on the Cartesian product of two complete graphs, and also on the hypercube case.

MSC:
05C57 Games on graphs (graph-theoretic aspects)
05C15 Coloring of graphs and hypergraphs
05B15 Orthogonal arrays, Latin squares, Room squares
05C76 Graph operations (line graphs, products, etc.)
91A43 Games involving graphs
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