On regular configurations and disjoint cycles in shift graphs

TL;DR

This paper explores the properties of regular configurations, their construction, symmetry, and applications to cycle packing in shift graphs, linking combinatorial structures with graph theoretical bounds.

Contribution

It introduces new insights into regular configurations, their uniqueness, symmetry, and their application to lower bounds in shift graph cycle packing.

Findings

Regular configurations are uniquely characterized and constructed.

A connection between regular configurations and balanced words is established.

Lower bounds for cycle packing numbers in shift graphs are derived.

Abstract

Configurations are necklaces with prescribed numbers of red and black beads. Among all possible configurations, the regular one plays an important role in many applications. In this paper, several aspects of regular configurations are discussed, including construction, uniqueness, symmetry group and the link with balanced words. Another model of configurations is the polygons formed by a given number of sides of two different lengths. In this context, regular configurations are used to obtain a lower bound for the cycles packing number of shift graphs, a subclass of the directed circulant graphs.