Dimension of unicycle posets

TL;DR

This paper proves a longstanding conjecture that finite posets with cover graphs containing at most one cycle have an order dimension of at most 3, providing explicit constructions for such posets.

Contribution

It offers a constructive proof confirming the conjecture and explicitly constructs triplets of linear extensions for these posets.

Findings

Confirmed the conjecture for posets with at most one cycle in cover graph

Provided explicit triplets of linear extensions for such posets

Established a constructive method for realizing the order dimension bound

Abstract

Motivated by the study of the dimension of random posets, it was conjectured by Bollob\'as and Brightwell in 1997 that if $P$ is a finite poset whose cover graph contains at most one cycle then its order dimension is at most $3$. In this paper we prove this conjecture by giving a constructive proof with explicit triplets of linear extensions realizing such posets.