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arXiv · 1612.01965

Packing Directed and Hamilton Cycles Online

Michael Anastos·Joseph Briggs
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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.

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Michael Anastos, Joseph Briggs. 2016-12-06. Packing Directed and Hamilton Cycles Online. https://arxiv.org/abs/1612.01965

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