Packing Directed and Hamilton Cycles Online

TL;DR

This paper demonstrates online algorithms for coloring edges in random directed and undirected graph processes to ensure the presence of Hamilton cycles in each color, achieving this with high probability.

Contribution

It introduces online coloring algorithms that guarantee Hamilton cycles in each color in random graph processes, a novel approach for online graph coloring.

Findings

Online coloring yields Hamilton cycles with high probability.

Algorithms work for both directed and undirected random graph processes.

Achieves multiple disjoint rainbow Hamilton cycles in the first suitable graph.

Abstract

Consider a directed analogue of the random graph process on $n$ vertices, where the $n(n-1)$ edges are ordered uniformly at random and revealed one at a time. It is known that w.h.p.\@ the first digraph in this process with both in-degree and out-degree $\geq q$ has a $[q]$-edge-coloring with a Hamilton cycle in each color. We show that this coloring can be constructed online, where each edge must be irrevocably colored as soon as it appears. In a similar fashion, for the \emph{undirected} random graph process, we present an online $[n]$-edge-coloring algorithm which yields w.h.p.\@ $q$ disjoint rainbow Hamilton cycles in the first graph of the process that contains $q$ disjoint Hamilton cycles.