Packing Hamilton Cycles Online

TL;DR

This paper proves that in an online setting, as edges are revealed one by one, it is possible to color them with  colors so that each color class forms a Hamilton cycle at the hitting time for minimum degree 2, extending known static results.

Contribution

It establishes an online coloring strategy that ensures  edge-disjoint Hamilton cycles appear simultaneously at the hitting time for minimum degree 2 in a random graph process.

Findings

Online coloring with  colors achieves Hamilton cycles at the hitting time.

The result extends static Hamilton cycle packing to an online setting.

High probability of success in the online process.

Abstract

It is known that w.h.p. the hitting time $\tau_{2\sigma}$ for the random graph process to have minimum degree $2\sigma$ coincides with the hitting time for $\sigma$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $\sigma\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $\sigma$ colors so that at time $\tau_{2\sigma}$, each color class is Hamiltonian.