Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rainbow perfect matchings and Hamilton cycles in the random geometric
  graph   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.

6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat