On degree sequences forcing the square of a Hamilton cycle

TL;DR

This paper proves a degree sequence condition that guarantees the existence of the square of a Hamilton cycle in large graphs, extending Pósa's conjecture with an asymptotically optimal criterion.

Contribution

It establishes a new degree sequence condition for Hamilton cycle squares, combining the Regularity-Blow-up and Connecting-Absorbing methods, and confirms the condition's asymptotic optimality.

Findings

Degree sequence condition guarantees Hamilton cycle square in large graphs

The condition is asymptotically best possible

Uses a hybrid of Regularity-Blow-up and Connecting-Absorbing techniques

Abstract

A famous conjecture of P\'osa from 1962 asserts that every graph on $n$ vertices and with minimum degree at least $2n/3$ contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Koml\'os, S\'ark\"ozy and Szemer\'edi. In this paper we prove a degree sequence version of P\'osa's conjecture: Given any $\eta >0$, every graph $G$ of sufficiently large order $n$ contains the square of a Hamilton cycle if its degree sequence $d_1\leq \dots \leq d_n$ satisfies $d_i \geq (1/3+\eta)n+i$ for all $i \leq n/3$. The degree sequence condition here is asymptotically best possible. Our approach uses a hybrid of the Regularity-Blow-up method and the Connecting-Absorbing method.