Hamilton cycles in graphs and hypergraphs: an extremal perspective

TL;DR

This survey reviews recent advances in Hamilton cycle research, emphasizing extremal, probabilistic, and hypergraph perspectives, and discusses open problems and new concepts like resilience and robustness.

Contribution

It provides a comprehensive overview of recent progress and open challenges in Hamilton cycle theory from an extremal and probabilistic viewpoint.

Findings

Resolved several long-standing problems using expansion and quasi-randomness

Highlighted the importance of resilience and robustness in Hamilton cycle studies

Explored Hamilton cycles in hypergraphs and their extremal properties

Abstract

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.