  In this paper, we propose a column by column encoding scheme suitable for
two-dimensional (2D) constraint codes and derive a lower bound of its maximum
achievable rate. It is shown that the maximum achievable rate is equal to the
largest minimum degree of a subgraph of the maximal valid pair graph. A graph
theoretical analysis to provide a lower bound of the maximum achievable rate is
presented. For several 2D-constraints such as the asymmetric and symmetric
non-isolated bit constraints, the values of the lower bound are evaluated.
