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

Abstract

Vizing's conjecture, dating back to 1963, asserts that \[ γ(G\mathbin{\square}H) \geq γ(G)γ(H) \] 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 \[ γ(G\mathbin{\square}H) \geq \frac{1}{2}γ(G)γ(H). \] Recently, Steiner obtained the first constant-factor improvement of the Clark--Suen bound, proving that \[ γ(G\mathbin{\square}H) \geq \frac{5+\sqrt{73}}{24}γ(G)γ(H) \approx 0.5643\,γ(G)γ(H). \] In this paper, we further improve the universal constant by proving that \[ γ(G\mathbin{\square}H) \geq \frac{2}{3}γ(G)γ(H) \] 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 $2/3$.

We fixed the argument and made an improvement. 10 pages