A constant-factor step towards Vizing's conjecture

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 $$γ(G\square H)\ge γ(G)γ(H),$$ where $γ$ 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 $γ(G\square H)\ge \frac{1}{2}γ(G)γ(H)$ 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 $γ(G\square H)\ge cγ(G)γ(H)$ 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 $$γ(G\square H)\ge cγ(G)γ(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx 0.5643.$$ Along the way, we prove another lower bound on $γ(G\square H)$ which outperforms the above bound for many graphs.