Getting directed Hamilton cycle twice faster

TL;DR

This paper demonstrates that by choosing edge orientations adaptively during a random graph process, one can achieve a directed Hamilton cycle as early as the undirected case, effectively halving the usual time threshold.

Contribution

It introduces an online orientation strategy that ensures the emergence of a directed Hamilton cycle simultaneously with the undirected graph's Hamiltonicity.

Findings

Directed Hamilton cycle appears at the same time as the undirected case.

Adaptive edge orientation can accelerate directed Hamiltonicity.

The result matches the undirected Hamiltonian threshold timing.

Abstract

Consider the random graph process where we start with an empty graph on n vertices, and at time t, are given an edge e_t chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph has a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if at time t, instead of a random direction one is allowed to choose the orientation of e_t, then whether it is possible or not to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p., the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.