4,088 research outputs found
On the threshold for rainbow connection number r in random graphs
We call an edge colouring of a graph G a rainbow colouring if every pair of
vertices is joined by a rainbow path, i.e., a path where no two edges have the
same colour. The minimum number of colours required for a rainbow colouring of
the edges of G is called the rainbow connection number (or rainbow
connectivity) rc(G) of G. We investigate sharp thresholds in the
Erd\H{o}s-R\'enyi random graph for the property "rc(G) <= r" where r is a fixed
integer. It is known that for r=2, rainbow connection number 2 and diameter 2
happen essentially at the same time in random graphs. For r >= 3, we conjecture
that this is not the case, propose an alternative threshold, and prove that
this is an upper bound for the threshold for rainbow connection number r.Comment: 16 pages, 2 figure
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
Rainbow -connectivity of random bipartite graphs
A path in an edge-colored graph is called a rainbow path if no two edges
of the path are colored the same. The minimum number of colors required to
color the edges of such that every pair of vertices are connected by at
least internally vertex-disjoint rainbow paths is called the rainbow
-connectivity of the graph , denoted by . For the random graph
, He and Liang got a sharp threshold function for the property
. In this paper, we extend this result to the case of
random bipartite graph .Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.1942 by
other author
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
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