The paper contains two main results. First, we obtain the chromatic polynomial on the \(n \times m\) section of the square lattice, solving a problem proposed by Read and Tutte \([5]\), the chromatic polynomial of the bracelet square lattice, and we find a recurrent-constructive process for the matrices of the \(k\)-colourings. The key concept for obtaining the inductive method is the compatible matrix.
Our second main result deals with the compatible matrix as the adjacency matrix of a graph. This represents a family of graphs, which is described.
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