1,069 research outputs found
The hitting time of rainbow connection number two
In a graph with a given edge colouring, a rainbow path is a path all of
whose edges have distinct colours. The minimum number of colours required to
colour the edges of so that every pair of vertices is joined by at least
one rainbow path is called the rainbow connection number of the graph
. For any graph , . We will show that for the
Erd\H{o}s-R\'enyi random graph close to the diameter 2 threshold, with
high probability if then . In fact, further strengthening
this result, we will show that in the random graph process, with high
probability the hitting times of diameter 2 and of rainbow connection number 2
coincide.Comment: 16 pages, 2 figure
Rainbow Connection of Random Regular Graphs
An edge colored graph is rainbow edge connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection of
a connected graph , denoted by , is the smallest number of colors
that are needed in order to make rainbow connected.
In this work we study the rainbow connection of the random -regular graph
of order , where is a constant. We prove that with
probability tending to one as goes to infinity the rainbow connection of
satisfies , which is best possible up to a hidden
constant
On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels
The q-composite key predistribution scheme [1] is used prevalently for secure
communications in large-scale wireless sensor networks (WSNs). Prior work
[2]-[4] explores topological properties of WSNs employing the q-composite
scheme for q = 1 with unreliable communication links modeled as independent
on/off channels. In this paper, we investigate topological properties related
to the node degree in WSNs operating under the q-composite scheme and the
on/off channel model. Our results apply to general q and are stronger than
those reported for the node degree in prior work even for the case of q being
1. Specifically, we show that the number of nodes with certain degree
asymptotically converges in distribution to a Poisson random variable, present
the asymptotic probability distribution for the minimum degree of the network,
and establish the asymptotically exact probability for the property that the
minimum degree is at least an arbitrary value. Numerical experiments confirm
the validity of our analytical findings.Comment: Best Student Paper Finalist in IEEE International Symposium on
Information Theory (ISIT) 201
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 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 dâdimensional cube for any fixed . 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 norm, for any fixed ). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of colours, where 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
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