General polygonal line tilings and their matching complexes

TL;DR

This paper extends previous work on the homotopy types of matching complexes of polygonal line tilings, analyzing new graph families and revealing Fibonacci number connections in specific cases.

Contribution

It broadens the class of polygonal line tilings studied and provides detailed analysis of triangle and pentagon lines, uncovering Fibonacci number patterns.

Findings

Matching complexes of extended polygonal line tilings are homotopy equivalent to wedges of spheres.

Fibonacci numbers appear in the analysis of triangle and pentagon line tilings.

The work generalizes previous results to larger graph families.

Abstract

A (general) polygonal line tiling is a graph formed by a string of cycles, each intersecting the previous at an edge, no three intersecting. In 2022, Matsushita proved the matching complex of a certain type of polygonal line tiling with even cycles is homotopy equivalent to a wedge of spheres. In this paper, we extend Matsushita's work to include a larger family of graphs and carry out a closer analysis of lines of triangle and pentagons, where the Fibonacci numbers arise.