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arXiv · 2606.14414

A constant-factor step towards Vizing's conjecture

Raphael Steiner
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Abstract

Vizing's conjecture from 1963, considered by many the most important open problem in the field of graph domination, states that all graphs $G$ and $H$ satisfy $$\gamma(G\square H)\ge \gamma(G)\gamma(H),$$ where $\gamma$ denotes the domination number and $\square$ the Cartesian product. In a seminal result, Clark and Suen (2000) proved an approximate form of the conjecture, namely that $\gamma(G\square H)\ge \frac{1}{2}\gamma(G)\gamma(H)$ for all graphs $G$ and $H$. Despite several lower-order improvements of this bound and improvements for special classes of graphs $G$ and $H$, no absolute constant $c>\frac{1}{2}$ such that $\gamma(G\square H)\ge c\gamma(G)\gamma(H)$ for all graphs $G$ and $H$, has been known thus far. In this paper, we obtain the first constant-factor improvement of the Clark-Suen bound by proving that for all graphs $G$ and $H$, we have $$\gamma(G\square H)\ge c\gamma(G)\gamma(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx 0.5643.$$ Along the way, we prove another lower bound on $\gamma(G\square H)$ which outperforms the above bound for many graphs.

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Raphael Steiner. 2026-06-12. A constant-factor step towards Vizing's conjecture. https://arxiv.org/abs/2606.14414

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