Symmetric sequencings and other combinatorial properties of large groups

TL;DR

The paper addresses the asymptotic theory of sequenceability and related sequencing properties for finite groups, proving that Anderson's conjecture on symmetric sequenceability and Bailey's conjecture on $2$-sequencings hold for sufficiently large groups. It develops a framework of derived sequences and leverages rainbow Hamilton sequences and Müesser–Pokrovskiy results to extend known properties (DR-, $R$-, $2$-, and harmonious sequences) to large abelian and non-abelian groups, including supersequenceability. A key outcome is that all abelian groups except $\mathbb{Z}_2$ are DR-sequenceable, and every sufficiently large group not elementary $2$-group is $2$-sequenceable and symmetric sequenceable, among other corollaries. The paper also analyzes partial sequencings and double sequencings, establishing extendability criteria and showing that large groups satisfy doubly sequenceable, doubly $R$-sequenceable, doubly harmonious, and doubly $R$-harmonious properties, thereby resolving several conjectures in the large-group regime.

Abstract

We prove that Anderson's conjecture on symmetric sequencings and Bailey's conjecture on 2-sequencings hold for sufficiently large groups. In addition, we discuss extensions of partial harmonious sequences and partial R-sequencings. Several further results on double sequencings are presented, both in the context of abelian groups and for sufficiently large non-abelian groups.

Theorems & Definitions (24)

• Definition 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Definition 5
• Theorem 6
• Corollary 7