The Queue-Number of Posets of Bounded Width or Height

TL;DR

This paper investigates the queue-number of posets, disproving a conjecture about planar posets' queue-number being bounded by height, and establishing bounds for posets of various widths, advancing understanding of their structural properties.

Contribution

It refutes a conjecture that planar posets' queue-number is bounded by height and provides new bounds for posets of different widths, including the first non-trivial case.

Findings

Planar posets can have queue-number larger than their height.

Posets of width 2 have queue-number at most 2.

Planar posets of width w have queue-number at most 3w-2.

Abstract

Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width $2$ has queue-number at most $2$, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width $w$ have queue-number at most $3w-2$ while any planar poset with $0$ and $1$ has queue-number at most its width.