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arXiv · 2309.00757

A Note on Hamiltonian-Intersecting Families of Graphs

Imre Leader·Žarko Ranđelović·Ta Sheng Tan
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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.

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Imre Leader, Žarko Ranđelović, Ta Sheng Tan. 2023-09-01. A Note on Hamiltonian-Intersecting Families of Graphs. https://arxiv.org/abs/2309.00757

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