On the Queue-Number of Partial Orders

TL;DR

This paper constructs posets with width w that have queue-numbers proportional to w squared, matching the known upper bounds and advancing understanding of the queue-number in relation to poset width.

Contribution

It provides the first construction of posets with width w having queue-number on the order of w squared, significantly improving previous lower bounds.

Findings

Constructed posets with queue-number Ω(w^2)

Matched the known upper bound asymptotically

Advanced the understanding of queue-number bounds for posets

Abstract

The queue-number of a poset is the queue-number of its cover graph viewed as a directed acyclic graph, i.e., when the vertex order must be a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue-number at most $w$. Recently, Alam et al. constructed posets of width $w$ with queue-number $w+1$. Our contribution is a construction of posets with width $w$ with queue-number $\Omega(w^2)$. This asymptotically matches the known upper bound.