An improved constant for Vizing's conjecture

Abstract

For any graph $G=(V,E)$, a subset $S\subseteq V$ dominates $G$ if $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written ${\hat{I}}^3(G)$. The classical conjecture of V.G. Vizing states that ${\hat{I}}^3(G\square H)\ge {\hat{I}}^3(G){\hat{I}}^3(H)$ where $\square$ stands for the Cartesian product of graphs. In this paper, we apply well-known results to prove the Vizing-type inequality ${\hat{I}}^3(G\square H)\ge .5809{\hat{I}}^3(G){\hat{I}}^3(H)$.

Algebraic mistake