Robust Hamiltonicity of Dirac graphs

TL;DR

This paper demonstrates that Dirac graphs remain Hamiltonian under random edge deletion and certain game conditions, showing their robustness in maintaining Hamiltonicity.

Contribution

The paper extends Dirac's theorem by proving Dirac graphs are robustly Hamiltonian against random subgraphs and Maker-Breaker games, under a unified framework.

Findings

Random subgraphs of Dirac graphs with p ≥ C log n / n are almost surely Hamiltonian.

Maker can build a Hamiltonian subgraph in a Dirac graph if the bias b ≤ c n / log n.

Results are tight up to a constant factor.

Abstract

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac's theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability $p$, and prove that there exists a constant $C$ such that if $p \ge C \log n / n$, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a $(1:b)$ Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias $b$ is at most $cn /\log n$ for some absolute constant $c > 0$. Both of these results are tight up to a constant factor, and are proved under one general framework.