[Paper Review] The regularity method for graphs and digraphs

Amelia Taylor|arXiv (Cornell University)|Jun 25, 2014

Limits and Structures in Graph Theory10 references3 citations

TL;DR

This paper applies the regularity method to solve extremal problems in undirected and directed graphs, focusing on Hamilton cycles and perfect packings. It proves that robustly outexpanding digraphs with linear minimum semidegree contain every orientation of a Hamilton cycle, providing an approximate solution to Nash-Williams' conjecture and extending results on perfect $C_6$-packings and degree sequence conditions.

ABSTRACT

This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szeméredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that every sufficiently large oriented graph on $n$ vertices with minimum in- and outdegree at least $3n/8 +o(n)$ contains any orientation of a Hamilton cycle. We use Kelly's arguments to extend his result to any robustly expanding digraph of linear degree.

Motivation & Objective

• To develop and apply the regularity method to undirected and directed graphs for solving extremal problems.
• To establish sufficient minimum degree and degree sequence conditions ensuring Hamilton cycles in digraphs.
• To prove that robust outexpanders with linear minimum semidegree contain every orientation of a Hamilton cycle.
• To provide an approximate solution to Nash-Williams' conjecture on degree sequence conditions for Hamilton cycles in digraphs.
• To extend results on perfect packings, such as $C_6$-packings, using the regularity method.

Proposed method

• Uses Szemerédi's Regularity Lemma to partition large graphs into clusters with pseudorandom edge distribution.
• Applies the Key Lemma and Blow-up Lemma to embed specified subgraphs into regular clusters.
• Employs the concept of $(\varepsilon,d)$-superregularity to ensure balanced edge distribution between clusters.
• Introduces robust outexpansion as a structural property resilient to small perturbations, enabling Hamilton cycle embedding.
• Uses degree sequence conditions slightly stronger than those in Nash-Williams' conjecture to imply robust outexpansion.
• Proves that every sufficiently large robust $(\nu,\tau)$-outexpander with linear minimum semidegree contains every orientation of a Hamilton cycle.

Experimental results

Research questions

• RQ1 Under what minimum degree or degree sequence conditions does a digraph contain a Hamilton cycle?
• RQ2 Can the regularity method be used to prove the existence of all orientations of a Hamilton cycle in large digraphs?
• RQ3 Does robust outexpansion imply the existence of every orientation of a Hamilton cycle in a digraph?
• RQ4 How do the conditions in Nash-Williams' conjecture relate to robust outexpansion and Hamilton cycle existence?
• RQ5 What minimum semidegree condition guarantees a perfect $C_6$-packing in large graphs?

Key findings

• Every sufficiently large robust $(\nu,\tau)$-outexpander with minimum semidegree at least $\eta n$ contains every orientation of a Hamilton cycle.
• The bound $\delta^0(G) \geq (3/8 + \alpha)n$ guarantees every orientation of a Hamilton cycle in large oriented graphs, confirming a result of Kelly.
• The result implies that degree sequence conditions slightly stronger than those in Nash-Williams' conjecture imply robust outexpansion and thus Hamilton cycle existence.
• The paper provides a sufficient minimum degree condition for perfect $C_6$-packings in large graphs using the regularity method.
• The regularity method successfully extends to oriented graphs, enabling the embedding of arbitrary orientations of Hamilton cycles.
• The main result generalizes previous theorems, including those by Keevash, Kühn, and Osthus, to all orientations of Hamilton cycles.