A brief, simple proof of Vizing's conjecture

TL;DR

This paper provides a concise and straightforward proof of Vizing's conjecture, which relates the domination numbers of graphs and their Cartesian products, confirming a long-standing hypothesis in graph theory.

Contribution

The paper offers a simple and direct proof of Vizing's conjecture, advancing understanding of domination in graph Cartesian products.

Findings

Proof confirms Vizing's conjecture for all graphs.

Establishes a clear relationship between domination numbers of graphs and their Cartesian products.

Simplifies previous approaches to the conjecture.

Abstract

For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{dominates} $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\gamma(G)$. In 1963, V.G. Vizing conjectured that $\gamma(G \square H) \geq \gamma(G)\gamma(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we prove the conjecture.