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arXiv · 2407.00103

Counting the Number of Domatic Partition of a Graph

Saeid Alikhani·Davood Bakhshesh·Nima Ghanbari
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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.

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Saeid Alikhani, Davood Bakhshesh, Nima Ghanbari. 2024-06-27. Counting the Number of Domatic Partition of a Graph. https://arxiv.org/abs/2407.00103

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