Rainbow perfect matchings and Hamilton cycles in the random geometric graph

TL;DR

This paper proves that in randomly edge-coloured geometric graphs, the presence of rainbow perfect matchings and Hamilton cycles aligns with minimum degree thresholds, extending classical graph theory results to the geometric and rainbow setting.

Contribution

It establishes the threshold conditions for rainbow perfect matchings and Hamilton cycles in random geometric graphs with random edge colours, a novel extension of classical combinatorial results.

Findings

Rainbow perfect matchings appear at minimum degree 1

Rainbow Hamilton cycles appear at minimum degree 2

Results hold for any fixed dimension d ≥ 2

Abstract

Given a graph on $n$ vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length $n$ visiting each vertex once and with pairwise different colours on the edges. Similarly (for even $n$) a rainbow perfect matching is a collection of $n/2$ independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least $1$ (respectively, at least $2$). More precisely, consider $n$ points (i.e. vertices) chosen independently and uniformly at random from the unit $d$-dimensional cube for any fixed $d\ge2$. Form a sequence of graphs on these $n$ vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the $\ell_p$ norm, for any fixed $1<p\le\infty$). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of $\lceil Kn\rceil$ colours, where $K=K(d)$ is a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least $1$ must contain a rainbow perfect matching (for even $n$), and the first graph with minimum degree at least $2$ must contain a rainbow Hamilton cycle.