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 $${\hat{I}}^3(G\square H)\ge {\hat{I}}^3(G){\hat{I}}^3(H),$$ where ${\hat{I}}^3$ 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 ${\hat{I}}^3(G\square H)\ge \frac{1}{2}{\hat{I}}^3(G){\hat{I}}^3(H)$ 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 ${\hat{I}}^3(G\square H)\ge c{\hat{I}}^3(G){\hat{I}}^3(H)$ 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 $${\hat{I}}^3(G\square H)\ge c{\hat{I}}^3(G){\hat{I}}^3(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx \mathrm{0.5643.}$$ Along the way, we prove another lower bound on ${\hat{I}}^3(G\square H)$ which outperforms the above bound for many graphs.

10 pages