Polyominoes and graphs built from Fibonacci words

TL;DR

This paper introduces and analyzes $k$-bonacci polyominoes and graphs, exploring their recursive structures, enumerations, and properties related to area, perimeter, and Hamiltonian cycles, connecting combinatorics and graph theory.

Contribution

It defines $k$-bonacci polyominoes and graphs, studies their recursive structures, and derives generating functions for various combinatorial properties, advancing understanding of these new objects.

Findings

Recursive structure of $k$-bonacci polyominoes determined

Enumeration formulas for area, perimeter, and word length derived

Generating functions for vertices, edges, and Hamiltonian cycles obtained

Abstract

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.