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      Two inequalities related to Vizing's conjecture

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          Abstract

          A well-known conjecture of Vizing is that \(\gamma(G \square H) \ge \gamma(G)\gamma(H)\) for any pair of graphs \(G, H\), where \(\gamma\) is the domination number and \(G \square H\) is the Cartesian product of \(G\) and \(H\). Suen and Tarr, improving a result of Clark and Suen, showed \(\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\min(\gamma(G),\gamma(H))\). We further improve their result by showing \(\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\max(\gamma(G),\gamma(H)).\) We also prove a fractional version of Vizing's conjecture: \(\gamma(G \square H) \ge \gamma(G)\gamma^*(H)\).

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          Most cited references 2

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          Vizing's conjecture: a survey and recent results

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            On the domination of the products of graphs II: Trees

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              Author and article information

              Journal
              2017-06-12
              Article
              1706.03682

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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              math.CO

              Combinatorics

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