Vizing猜想的一个改进常数
A $2/3$ Bound for Vizing's Conjecture
浏览论文内容
中文总结 AI 辅助
本文利用已知结果证明Vizing型不等式γ(G□H) ≥ 0.5809 γ(G)γ(H)，改进了Vizing猜想的常数下界。
中文摘要 AI 辅助
对于任意图$G = (V,E)$，子集$S {\subseteq} V$支配$G$如果$N[S] = V$。所有这样的$S$的最小基数称为支配数，记作${\gamma}(G)$。V.G. Vizing的经典猜想指出${\gamma}(G{\square} H) {\ge} {\gamma}(G){\gamma}(H)$，其中${\square}$表示图的笛卡尔积。在本文中，我们应用已知结果证明Vizing型不等式${\gamma}(G{\square} H) {\ge}.5809 {\gamma}(G){\gamma}(H)$。
英文摘要
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 $γ$ 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 $2/3$.
发表机构
- Clayton State University（克莱顿州立大学）
机构由 AI 辅助整理，请以论文原文为准。
