Families of locally separated Hamilton paths

TL;DR

This paper presents an explicit construction that significantly improves the lower bounds on the size of families of Hamilton paths with specific union properties in complete graphs, also extending results to permutations.

Contribution

It provides an explicit construction for large families of Hamilton paths with union degree constraints, surpassing previous bounds obtained by greedy algorithms.

Findings

Exponential improvement in lower bounds for Hamilton path families.

Explicit construction method demonstrated.

Results extend to permutation families.

Abstract

We improve by an exponential factor the lower bound of Korner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor.