Families of graph-different Hamilton paths

TL;DR

This paper investigates the maximum size of sets of Hamiltonian paths in complete graphs where unions contain cycles of specified lengths, extending classical graph intersection and permutation capacity problems.

Contribution

It determines asymptotic bounds for these sets in various cases and introduces cycle-difference generalizations, connecting to Shannon capacity and graph intersection theories.

Findings

Established bounds for M(n;D) in special cases

Linked cycle-difference problems to permutation capacity

Presented generalizations involving cliques

Abstract

Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K_n such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M(n;D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erd\"os, Simonovits and S\'os. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging.