Counting Colored Tilings on Grids and Graphs

TL;DR

This paper investigates generalized tiling counting problems on grid-like structures, providing explicit generating functions for specific cases, extending previous combinatorial results to graph products.

Contribution

It introduces new methods for counting tilings on graph products and derives explicit generating functions for certain configurations.

Findings

Derived explicit bivariate generating functions for specific graph tiling cases

Extended classical grid tiling counting to product graphs

Provided combinatorial formulas for tilings on G x P(n) structures

Abstract

In this paper, we explore some generalizations of a counting problem related to tilings in grids of size 2xn, which was originally posed as a question on Mathematics Stack Exchange (Question 3972905). In particular, we consider this problem for the product of two graphs G and P(n), where P(n) is the path graph of n vertices. We give explicit bivariate generating functions for some specific cases.