On the total {k}-domination number of Cartesian products of graphs

Ning Li and Xinmin Hou ()
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Ning Li: University of Science and Technology of China
Xinmin Hou: University of Science and Technology of China

Journal of Combinatorial Optimization, 2009, vol. 18, issue 2, No 4, 173-178

Abstract: Abstract Let γ t {k} (G) denote the total {k}-domination number of graph G, and let $G\mathbin{\square}H$ denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, $\gamma _{t}^{\{k\}}(G)\gamma _{t}^{\{k\}}(H)\le k(k+1)\gamma _{t}^{\{k\}}(G\mathbin{\square}H)$ . As a corollary of this result, we have $\gamma _{t}(G)\gamma _{t}(H)\le 2\gamma _{t}(G\mathbin{\square}H)$ for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005).

Keywords: Total {k}-domination; Total domination; Cartesian product (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10878-008-9144-2

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