The orthogonal colouring game

We introduce the orthogonal colouring game, in which two players alternately colour vertices (from a choice of m is an element of N colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the colouring. Each player aims to maximise her score, which is the number of coloured vertices in the copy of the graph she owns. The main result of this paper is that the second player has a strategy to force a draw in this game for any m is an element of N for graphs that admit a strictly matched involution. An involution sigma of a graph G is strictly matched if its fixed point set induces a clique and any non-fixed point v is an element of V (G) is connected with its image sigma(v) by an edge. We give a structural characterisation of graphs admitting a strictly matched involution and bounds for the number of such graphs. Examples of such graphs are the graphs associated with Latin squares and sudoku squares. (C) 2019 Elsevier B.V. All rights reserved.