Enumeration of Colored Tilings on Graphs via Generating Functions

TL;DR

The paper develops a comprehensive generating-function framework to enumerate $k$-colored partitions (tilings) of graphs, focusing on how uniform random coloring induces a partition into connected blocks. It derives exact bivariate (and exponential) generating functions for several graph families—trees, cycles, complete graphs, complete bipartite graphs, and Cartesian products $K_m\times P_n$—and uses these to obtain closed-form expressions for the expected number of blocks under coloring. A unifying approach for Cartesian products employs last-slice decompositions and symmetry reductions to manage the complexity, including explicit results for $m=3,4$ and general constructions via linear-systems reductions. The work broadens classic tiling enumeration to colored partitions on graph products, yielding explicit formulas and generating functions with potential applications to polyomino tilings and related combinatorial structures. The formulas provide direct means to quantify expected tiling complexity under uniform colorings, with exact rational expressions and scalable methods for larger graph products.

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.

Figures (9)

• Figure 1: A tiling of a $4 \times 10$ grid using polyominoes.
• Figure 2: A 3-colored partition of size $11$ of $L_{4,10}$.
• Figure 3: The 2-colored partitions of size 3 for the perfect binary tree $B_2$.
• Figure 4: The 2-colored partitions of size 4 for the cycle graph $C_5$.
• Figure 5: The 2-colored partitions of size 2 for the complete graph $K_4$.

Theorems & Definitions (30)

• Lemma 2.1
• proof
• Corollary 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Theorem 3.1
• proof
• Corollary 3.2