Vizing's Conjecture for Almost All Pairs of Graphs

TL;DR

This paper proves Vizing's conjecture for almost all pairs of graphs under certain size conditions, advancing understanding of domination numbers in Cartesian product graphs.

Contribution

It establishes that Vizing's conjecture holds for almost all pairs of graphs when their sizes meet specific bounds, a significant step forward in graph domination theory.

Findings

Vizing's conjecture holds if |G| and |H| are at least the product of their domination numbers.

The conjecture is valid for almost all pairs of graphs with sizes constrained by a probabilistic bound.

The result applies to graphs with sizes up to a certain exponential function of |H|, depending on the edge probability p.

Abstract

For any graph $G=(V,E)$, a subset $S\subseteq V$ $dominates$ $G$ 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 that if $\left|G\right|\geq \gamma(G)\gamma(H)$ and $\left|H\right|\geq \gamma(G)\gamma(H)$, then the conjecture holds. This result quickly implies Vizing's conjecture for almost all pairs of graphs $G,H$ with $\left|G\right|\geq \left|H\right|$, satisfying $\left|G\right|\leq q^{\frac{\left|H\right|}{\log_q\left|H\right|}}$ for $q=\frac{1}{1-p}$ and $p$ the edge probability of the Erd\H{o}s-R\'enyi random graph.