4 research outputs found
Power of choices and rainbow spanning trees in random graphs
We consider the Erd\H{o}s-R\'enyi random graph process, which is a stochastic
process that starts with vertices and no edges, and at each step adds one
new edge chosen uniformly at random from the set of missing edges. Let
be a graph with edges obtained after steps of this
process. Each edge () of independently
chooses precisely colours, uniformly at random, from a given
set of colours (one may view as a multi-edge). We stop the process
prematurely at time when the following two events hold:
is connected and every colour occurs at least once ( if some
colour does not occur before all edges are present; however, this does not
happen asymptotically almost surely). The question addressed in this paper is
whether has a rainbow spanning tree (that is, multicoloured
tree on vertices). Clearly, both properties are necessary for the desired
tree to exist.
In 1994, Frieze and McKay investigated the case and the answer to this
question is "yes" (asymptotically almost surely). However, since the sharp
threshold for connectivity is and the sharp threshold for
seeing all the colours is , the case is of special
importance as in this case the two processes keep up with one another. In this
paper, we show that asymptotically almost surely the answer is "yes" also for
Multicoloured Hamilton cycles in random graphs; an anti-Ramsey threshold
Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a k-bounded colouring. We say that a subset of the edges of G is multicoloured if each edge is of a different colour. We say that the colouring is H-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edgeset. Let AR k = fG : every k-bounded colouring of G is H-goodg. We establish the threshold for the random graph G n;m to be in AR k