The upper domatic number of powers of graphs

Document Type: Original paper

Authors

1 CHRIST (Deemed to be University), Bangalore

2 CHRIST(Deemed to be University), Bangalore

10.22049/cco.2020.26913.1163

Abstract

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 $k=2$. We also show that the upper domatic number of $k^{mathrm{th}}$ power of a graph can be viewed as the $ k$-upper domatic number of a graph.

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