Minimal Asymmetric Graphs

TL;DR

This paper proves that there are exactly 18 finite minimal asymmetric undirected graphs, confirming a conjecture and linking them to minimal involution-free graphs, thus advancing understanding of graph symmetries.

Contribution

It establishes the exact number of finite minimal asymmetric graphs and characterizes them as minimal involution-free graphs, confirming a longstanding conjecture.

Findings

Exactly 18 such graphs exist.

These graphs are precisely the minimal involution-free graphs.

The result confirms a conjecture of Nešetřil.

Abstract

Confirming a conjecture of Ne\v{s}et\v{r}il, we show that up to isomorphism there is only a finite number of finite minimal asymmetric undirected graphs. In fact, there are exactly 18 such graphs. We also show that these graphs are exactly the finite minimal involution-free graphs.