Hamilton cycles in sparse robustly expanding digraphs

TL;DR

This paper proves that sparse robustly expanding directed graphs contain Hamilton cycles without relying on Szemerédi's Regularity Lemma, extending results to sparser graphs.

Contribution

It provides a new proof demonstrating Hamilton cycles in sparse robustly expanding digraphs without using the Regularity Lemma, broadening applicability.

Findings

Sparse robustly expanding digraphs contain Hamilton cycles.

The proof avoids Szemerédi's Regularity Lemma.

Results apply to sparser graphs than previously possible.

Abstract

The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle. Previous proofs of this require Szemer\'edi's Regularity Lemma and so this fact can only be applied to dense, sufficiently large robust expanders. We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to suitable sparse robustly expanding digraphs.