A survey of the cell-growth problem and some its variations

A very brief survey of the main results concerning the cell–growth problem and its variations is given. The name stems from an analogy with an animal which, starting from a single cell of some specified basic polygonal shape, grows step by step in the plane by adding at each step a cell of the same shape to its periphery. The fundamental combinatorial problem concerning these animals is ”How many animals with n cells are there?” This problem was included in the list of unsolved problems in the enumeration of graphs by Frank Harary in 1960. Despite serious efforts over the last 40 years, this problem is completely open. However, a few asymptotic results are known. For example, let p(n) denote the number of polyominoes (square animals) having n cells. It was proved that (p(n)) tends to a limit Θ, which satisfies the following inequality: 3.87 < Θ < 4.65 . The situation could hardly be worse, since the first digit of Θ is not even known... The difficulty of the classical cell–growth problem has led to the study of various restricted classes of polyominoes. Some variations of this problem are considered. Unsolved problems are stated. Chemical applications of this problem are mentioned too. 1. Classical cell–growth problem Combinatorial problem known as cell-growth problem is stated as follows [1–7]. The name stems from an analogy with an animal which, starting from a single cell of some specified basic polygonal shape, grows step by step in the plane by adding at each step a cell of the same shape to its periphery. Thus if the basic shape is a square, the animals are the polyominoes (Fig.1a). If the basic shape is an equilateral triangle or a regular hexagon, we obtain triangular and hexagonal animals looking like those in Fig.1b and Fig.1c. Animals are defined as simply–connected ones if they have no holes and as multiply–connected ones otherwise. All animals presented in Fig.1 are simply–connected ones. The smallest multiply–connected polyomino is shown in Fig.2. The fundamental combinatorial problem concerning these animals is ”How many animals with n cells are there?” This problem was included in the list of unsolved problems in the enumeration of graphs by Harary in 1960 [8]. Polyominoes have the most long history, going to the start of the 20th century, but 1Supported by Com2MaC-KOSEF, The Republic of Korea.