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- W3138367664 abstract "A emph{domatic} (emph{total domatic}) emph{$k$-coloring} of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that each vertex contains vertices of all $k$ colors in its closed neighborhood (neighborhood). The emph{domatic} (emph{total domatic}) emph{number} of $G$, denoted $d(G)$ ($d_t (G)$), is the maximum $k$ for which $G$ has a domatic (total domatic) $k$-coloring. In this paper, we show that for two non-trivial graphs $G$ and $H$, the domatic and total domatic numbers of their Cartesian product $G cart H$ is bounded above by $max{|V(G)|, |V(H)|}$ and below by $max{d(G), d(H)}$. Both these bounds are tight for an infinite family of graphs. Further, we show that if $H$ is bipartite, then $d_t(G cart H)$ is bounded below by $2min{d_t(G),d_t(H)}$ and $d(G cart H)$ is bounded below by $2min{d(G),d_t(H)}$. These bounds give easy proofs for many of the known bounds on the domatic and total domatic numbers of hypercubes cite{chen,zel4} and the domination and total domination numbers of hypercubes cite{har,joh} and also give new bounds for Hamming graphs. We also obtain the domatic (total domatic) number and domination (total domination) number of $n$-dimensional torus $mathop{cart}limits_{i=1}^{n} C_{k_i}$ with some suitable conditions to each $k_i$, which turns out to be a generalization of a result due to Gravier cite{grav2} %[emph{Total domination number of grid graphs}, Discrete Appl. Math. 121 (2002) 119-128] and give easy proof of a result due to Klavv{z}ar and Seifter cite{sand}." @default.
- W3138367664 created "2021-03-29" @default.
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- W3138367664 date "2021-03-19" @default.
- W3138367664 modified "2023-09-27" @default.
- W3138367664 title "On Domatic and Total Domatic Numbers of Product Graphs" @default.
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- W3138367664 doi "https://doi.org/10.48550/arxiv.2103.10713" @default.
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