Sparseness of 4-cycle systems

TL;DR

This paper investigates the sparseness properties of 4-cycle systems, generalizing Erdős' r-sparse conjecture, and proves the existence of strictly r-sparse systems and packings for large orders.

Contribution

It introduces the concept of strict r-sparseness in 4-cycle systems and proves their existence for all admissible orders and large sizes.

Findings

Existence of strictly 4-sparse 4-cycle systems for all admissible orders.

For any r > 1, large order 4-cycle packings with bounded sparseness exist.

Construction methods for sparse 4-cycle systems are developed.

Abstract

An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erd\H{o}s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it contains no configurations consisting of j 4-cycles whose union contains precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a configuration on two 4-cycles sharing a diagonal, called the double-diamond, we say it is strictly r-sparse. In this paper, we show that for every admissible order v there exists a strictly 4-sparse 4CS(v). We also prove that for any positive integer r > 1 and sufficiently large integer v there exists a constant number c such that there exists a strictly r-sparse 4-cycle packing of order v with cv^2 4-cycles.