For the maximum number f ( n ) of edges in a C4-free subgraph of the n-dimensional cube-graph 0, w e prove f(n) 2 i ( n + f i ) 2 " -' for n = 4f, and f ( n ) 2 i ( n + 0.9,h)2"-' for all n 2 9. This disproves one version of a conjecture of P. Erdos. 0 1995 John Wiley & Sons, Inc.

Chess-like game boards B n are considered, which are hexagonal parts of the Euclidean tessellation of the plane by regular hexagons. For chess-like rooks on B n the domination number γ(n) is determined.