Enumeration of Colored Tilings on Graphs via Generating Functions

TL;DR

This paper develops a generating function approach to count and analyze colored tilings on graphs, especially Cartesian products of graphs with paths, generalizing grid tiling enumeration with applications in combinatorics.

Contribution

It introduces a novel combinatorial method using bivariate generating functions to enumerate colored graph partitions, extending tiling enumeration to broader graph families.

Findings

Derived formulas for expected number of blocks in colored graph partitions

Generalized grid tiling enumeration to graph Cartesian products

Provided combinatorial techniques applicable to various graph classes

Abstract

In this paper, we study the problem of partitioning a graph into connected and colored components called blocks. Using bivariate generating functions and combinatorial techniques, we determine the expected number of blocks when the vertices of a graph $G$, for $G$ in certain families of graphs, are colored uniformly and independently. Special emphasis is placed on graphs of the form $G \times P_n$, where $P_n$ is the path graph on $n$ vertices. This case serves as a generalization of the problem of enumerating the number of tilings of an $m \times n$ grid using colored polyominoes.