Counting the Number of Domatic Partition of a Graph

TL;DR

This paper introduces the domatic partition polynomial, a generating function for counting domatic partitions of a graph, and provides algorithms and results for trees, paths, and graph products.

Contribution

It defines the domatic partition polynomial, offers a quadratic time algorithm for trees, and explores its properties for specific graph classes.

Findings

Quadratic time algorithm for computing the domatic polynomial of trees.

Explicit formulas for paths and certain graph products.

Practical applications demonstrated through theoretical results.

Abstract

A subset of vertices $S$ of a graph $G$ is a dominating set if every vertex in $V \setminus S$ has at least one neighbor in $S$. A domatic partition is a partition of the vertices of a graph $G$ into disjoint dominating sets. The domatic number $d(G)$ is the maximum size of a domatic partition. Suppose that $dp(G,i)$ is the number of distinct domatic partition of $G$ with cardinality $i$. In this paper, we consider the generating function of $dp(G,i)$, i.e., $DP(G,x)=\sum_{i=1}^{d(G)}dp(G,i)x^i$ which we call it the domatic partition polynomial. We explore the domatic polynomial for trees, providing a quadratic time algorithm for its computation based on weak 2-coloring numbers. Our results include specific findings for paths and certain graph products, demonstrating practical applications of our theoretical framework.