Clumsy packing of dominoes

One places dominoes on ann x n chessboard. Each domino must cover exactly two squares of the board and no two dominoes may overlap. The board is said to be full if there is no room for placing further dominoes (Fig. 1). The problem of clumsy packing is formulated as follows: what is the minimum number d(n) of dominoes lying on ann X n full board? In fact, the clumsy packing of dominoes can be viewed as the worst case of the 'greedy' packing. That kind of domineering is known under various names as a two person game (Cram, Plugg, Dots and pairs). Either player may place his dominoes in either direction and the player who finds no room for his next domino, loses (c.f. [1]). For this game the obvious meaning of the function d(n) is that the game should not be finished in less than d(n) moves. Here we prove that d(n) = n /3 if 3 In and d(n) > n 2 /3 + n/111 if n is large and not divisible by 3. Let G be a graph. A matching of G is a set of independent edges. The ratio of the minimum and maximum size of a maximal matching, r(G), measures the worst case behaviour of the greedy matching algorithm. It was proved in [2] that r( G) ?: 1/2 for any graph G. If we consider the graph G(n) whose vertices are the squares of the board and whose edges correspond to squares having a common side, then the packing of dominoes corresponds to a matching, i.e. a set of independent edges of G(n ). The meaning of the function d(n) is obviously the minimum size of a maximal matching. The maximum size of a maximal matching of G(n) is about n /2, therefore, our result shows that r( G) is about 2/3 for chessboard graphs. In Section 3, the problem of clumsy packing, or equivalently, the ratio r( G) is