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Coloring the faces of 2-dimensional square lattice with $k$ distinct colorssuch that no two adjacent faces have the same color is considered byestablishing connection between the $k$ coloring problem and a generalizedvertex model. Associating the colors with $k$ distinct species of particleswith infinite repulsive force between nearest neighbors of the same type andzero chemical potential $\mu$ associated with each species, the number of ways$[W(k)]^N$ for large $N$ is related to the entropy of the {\it{hard squarelattice gas}} at close packing of the lattice, where $N$ is the number oflattice sites. We discuss the evaluation of $W(k)$ using transfer matrix methodwith non-periodic boundary conditions imposed on at least one dimension andshow the characteristic Toeplitz block structure of the transfer matrix. Usingthis result, we present some analytical calculations for non-periodic modelsthat remain finite in one dimension. The case $k=3$ is found to approach theexact result obtained by Lieb for the residual entropy of ice with periodicboundary conditions. Finally, we show, by explicit calculation of thecontribution of subgraphs and the series expansion of $W(k)$, that thegenenralized Pauling type estimate(which is based on mean field approximation)dominates at large values of $k$. We thus also provide an alternative seriesexpansion for the chromatic polynomial of a regular square graph.

Entropy of hard square lattice gas with $k$ distinct species of particles: coloring problems and vertex models