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Preprint

2017-06-12

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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Sandi Klavžar, and Rall, Boštjan Brešar … (2012)

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Lael Kinch, Michael Jacobson (1987)

1706.03682

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