Path separation by short cycles

TL;DR

This paper investigates the maximum size of Hamilton path families in complete graphs where each pair is separated by a cycle of fixed length, providing asymptotic bounds for small cycle lengths.

Contribution

It introduces bounds on the maximum number of Hamilton paths in $K_n$ with pairwise separation by a cycle of fixed length $k$, advancing understanding of cycle-based path separation.

Findings

Established asymptotic bounds for small fixed $k$

Characterized the maximum family size for Hamilton paths with cycle separation

Extended previous work on cycle and path combinatorics

Abstract

Two Hamilton paths in $K_n$ are separated by a cycle of length $k$ if their union contains such a cycle. For small fixed values of $k$ we bound the asymptotics of the maximum cardinality of a family of Hamilton paths in $K_n$ such that any pair of paths in the family is separated by a cycle of length $k.$