%0 Journal Article
%T The upper domatic number of powers of graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Samuel, Libin Chacko
%A Joseph, Mayamma
%D 2021
%\ 06/01/2021
%V 6
%N 1
%P 53-65
%! The upper domatic number of powers of graphs
%K Domatic number
%K $k$-domatic number
%K Upper domatic partition
%K Upper domatic number
%K $k$-upper domatic number
%R 10.22049/cco.2020.26913.1163
%X 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.
%U http://comb-opt.azaruniv.ac.ir/article_14108_7058eb1b6f8a087a8b1ec3b80f28c2ac.pdf