A semi-exact degree condition for Hamilton cycles in digraphs

TL;DR

This paper establishes new degree conditions that guarantee Hamiltonicity in large directed graphs, extending classical theorems and providing approximate solutions to longstanding conjectures.

Contribution

It introduces semi-exact degree conditions for Hamilton cycles in digraphs, weakening previous assumptions and advancing the understanding of Hamiltonicity criteria.

Findings

Proves Hamiltonicity under degree conditions involving min{i + a n, n/2}.

Weakens assumptions compared to previous theorems.

Provides an approximate version of Nash-Williams' conjecture.

Abstract

The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and Treglown.