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arXiv · 2607.01109

A $2/3$ Bound for Vizing's Conjecture

Mohsen Aliabadi·Elliot Krop
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Abstract

Vizing's conjecture, dating back to 1963, asserts that \[ \gamma(G\mathbin{\square}H) \geq \gamma(G)\gamma(H) \] for all finite graphs $G$ and $H$, where $\gamma$ denotes the domination number and $\square$ denotes the Cartesian product. In 2000, Clark and Suen proved the universal bound \[ \gamma(G\mathbin{\square}H) \geq \frac{1}{2}\gamma(G)\gamma(H). \] Recently, Steiner obtained the first constant-factor improvement of the Clark--Suen bound, proving that \[ \gamma(G\mathbin{\square}H) \geq \frac{5+\sqrt{73}}{24}\gamma(G)\gamma(H) \approx 0.5643\,\gamma(G)\gamma(H). \] In this paper, we further improve the universal constant by proving that \[ \gamma(G\mathbin{\square}H) \geq \frac{2}{3}\gamma(G)\gamma(H) \] for all finite graphs $G$ and $H$. Thus, we raise the best known universal constant in the approximate form of Vizing's conjecture from $(5+\sqrt{73})/24$ to $2/3$.

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Mohsen Aliabadi, Elliot Krop. 2026-07-01. A $2/3$ Bound for Vizing's Conjecture. https://arxiv.org/abs/2607.01109

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