Counting the Number of Domatic Partition of a Graph
Saeid Alikhani, Davood Bakhshesh, Nima Ghanbari
TL;DR
The paper studies the domatic partition polynomial $DP(G,x) = \sum_{i=1}^{d(G)} dp(G,i) x^i$ that counts domatic partitions by size, where $dp(G,i)=|DP(G,i)|$. It establishes basic bounds $d(G)\le \delta+1$, links $dp(G,2)$ to the weak $2$-colorings count $w_2(G)$, and notes that isolates force $DP(G,x)=x$, with $dp(G,1)=1$ and derivatives $\frac{d^r}{dx^r} DP(G,x)=r!\, dp(G,r)$. Computing $DP$ is NP-complete, motivating focused analysis on trees. For trees, the paper proves $d(T)=2$ and $DP(T,x)= x + w_2(T) x^2$, and develops a quadratic-time algorithm to compute $w_2(T)$ using a weak $2$-coloring framework with recursive decompositions around star-neighbor structures; it also provides explicit results for paths and certain graph products, such as $DP(P_n\circ K_1,x)=x+2^{n-1}x^2$ and $DP(G\circ \overline{K}_r,x)=x+2^{n-1}x^2$. Together, these contributions offer a practical framework for enumerating domatic partitions and highlight both complexity and tractable special cases in graph theory.
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.