Polyominoes and graphs built from Fibonacci words

11/10/2022

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by   Sergey Kirgizov, et al.

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Universidad Nacional de Colombia

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Université de Bourgogne

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We introduce the k-bonacci polyominoes, a new family of polyominoes associated with the binary words avoiding k consecutive 1's, also called generalized k-bonacci words. The polyominoes are very entrancing objects, considered in combinatorics and computer science. The study of polyominoes generates a rich source of combinatorial ideas. In this paper we study some properties of k-bonacci polyominoes. Specifically, we determine their recursive structure and, using this structure, we enumerate them according to their area, semiperimeter, and length of the corresponding words. We also introduce the k-bonacci graphs, then we obtain the generating functions for the total number of vertices and edges, the distribution of the degrees, and the total number of k-bonacci graphs that have a Hamiltonian cycle.