The upper domatic number of powers of graphs

Document Type : Original paper


1 CHRIST (Deemed to be University), Bangalore

2 CHRIST(Deemed to be University), Bangalore


Let $A$ and $B$ be two disjoint subsets of the vertex set $V$ of a graph $G$. The set $A$ is said to dominate $B$, denoted by $A \rightarrow B$, if for every vertex $u \in B$ there exists a vertex $v \in A$ such that $uv \in E(G)$. For any graph $G$, a partition $\pi = \{V_1,$ $V_2,$ $\ldots,$ $V_p\}$ of the vertex set $V$ is an \textit{upper domatic partition} if $V_i \rightarrow V_j$ or $V_j \rightarrow V_i$ or both for every $V_i, V_j \in \pi$, whenever $i \neq j$. The \textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. In this paper, we study the upper domatic number of powers of graphs and examine the special case when power is $2$. We also show that the upper domatic number of $k^{\mathrm{th}}$ power of a graph can be viewed as its $ k$-upper domatic number.


Main Subjects

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