396 research outputs found
Unicyclic Components in Random Graphs
The distribution of unicyclic components in a random graph is obtained
analytically. The number of unicyclic components of a given size approaches a
self-similar form in the vicinity of the gelation transition. At the gelation
point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a
result, the total number of unicyclic components grows logarithmically with the
system size.Comment: 4 pages, 2 figure
On the probability of planarity of a random graph near the critical point
Consider the uniform random graph with vertices and edges.
Erd\H{o}s and R\'enyi (1960) conjectured that the limit
\lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists
and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994)
proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower
and upper bounds for this probability.
In this paper we determine the exact probability of a random graph being
planar near the critical point . For each , we find an exact
analytic expression for
In particular, we obtain .
We extend these results to classes of graphs closed under taking minors. As
an example, we show that the probability of being
series-parallel converges to 0.98003.
For the sake of completeness and exposition we reprove in a concise way
several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur
Creation and Growth of Components in a Random Hypergraph Process
Denote by an -component a connected -uniform hypergraph with
edges and vertices. We prove that the expected number of
creations of -component during a random hypergraph process tends to 1 as
and tend to with the total number of vertices such that
. Under the same conditions, we also show that
the expected number of vertices that ever belong to an -component is
approximately . As an immediate
consequence, it follows that with high probability the largest -component
during the process is of size . Our results
give insight about the size of giant components inside the phase transition of
random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend
On the length of a random minimum spanning tree
We study the expected value of the length of the minimum spanning tree
of the complete graph when each edge is given an independent uniform
edge weight. We sharpen the result of Frieze \cite{F1} that
\lim_{n\to\infty}\E(L_n)=\z(3) and show that
\E(L_n)=\z(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}} where are
explicitly defined constants.Comment: Added next term and two co-author
Forbidden Subgraphs in Connected Graphs
Given a set of connected non acyclic graphs, a
-free graph is one which does not contain any member of as copy.
Define the excess of a graph as the difference between its number of edges and
its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating
function (EGF for brief) of connected -free graphs of excess equal to
(). For each fixed , a fundamental differential recurrence
satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to
solve this nonlinear recurrence for the first few values of by means of
graph surgery. We also show that for any finite collection of non-acyclic
graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the
generating function, , of Cayley's rooted (non-planar) labelled trees. From
this, we prove that almost all connected graphs with nodes and edges
are -free, whenever and by means of
Wright's inequalities and saddle point method. Limiting distributions are
derived for sparse connected -free components that are present when a
random graph on nodes has approximately edges. In particular,
the probability distribution that it consists of trees, unicyclic components,
, -cyclic components all -free is derived. Similar results are
also obtained for multigraphs, which are graphs where self-loops and
multiple-edges are allowed
On-line list colouring of random graphs
In this paper, the on-line list colouring of binomial random graphs G(n,p) is
studied. We show that the on-line choice number of G(n,p) is asymptotically
almost surely asymptotic to the chromatic number of G(n,p), provided that the
average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log
n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that
if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is
larger than the chromatic number by at most a multiplicative factor of C, where
C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice
number is by at most a multiplicative constant factor larger than the chromatic
number
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