A Note on Hamiltonian-Intersecting Families of Graphs

TL;DR

This paper extends a known result about intersecting families of graphs to directed graphs and Hamiltonian intersections, establishing upper bounds and confirming a conjecture for Hamiltonian intersecting families.

Contribution

It provides a directed version of the intersecting graph family result and verifies a conjecture regarding Hamiltonian intersection sizes.

Findings

Directed strongly-connected intersection family size at most 1/3^n of all oriented graphs.

Hamiltonian intersection family size at most 1/2^n of all graphs.

Confirmation of a conjecture on Hamiltonian intersecting families.

Abstract

How many graphs on an $n$-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly $1/2^{n-1}$ of all graphs. Our aim in this short note is to give a 'directed' version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most $1/3^n$ of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most $1/2^n$ of all graphs, verifying a conjecture of the above authors.