Circular game chromatic number of graphs

In a circular r -colouring game on G , Alice and Bob take turns colouring the vertices of G with colours from the circle S ( r ) of perimeter r . Colours assigned to adjacent vertices need to have distance at least 1 in S ( r ) . Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χ c g ( G ) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r -colouring game on G . This paper proves that for any graph G , χ c g ( G ) ≤ 2 col g ( G ) − 2 , where col g ( G ) is the game colouring number of G . This upper bound is shown to be sharp for forests. It is also shown that for any graph G , χ c g ( G ) ≤ 2 χ a ( G ) ( χ a ( G ) + 1 ) , where χ a ( G ) is the acyclic chromatic number of G . We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles.