9 research outputs found
Lower Bounds on the Chromatic Number of Random Graphs
We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborová and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we obtain a variational formula. As an application we calculate improved explicit lower bounds on the chromatic number of random graphs for small (average) degrees. Additionally, we show how asymptotic formulas for large degrees that were previously obtained by lengthy and complicated combinatorial arguments can be re-derived easily from these new results
Properties of random graphs
The thesis describes new results for several problems in random graph theory.
The first problem relates to the uniform random graph model in
the supercritical phase; i.e. a graph, uniformly distributed, on vertices
and edges for satisfying
and . The property studied is the length of the
longest cycle in the graph. We give a new upper bound, which holds
asymptotically almost surely, on this length.
As part of our proof we establish a result about the heaviest cycle in a certain
randomly-edge-weighted nearly-3-regular graph, which may be of independent interest.
Our second result is a new contiguity result for a random -regular
graph. Let be a function that is linear in .
A -irregular graph is a graph which is -regular except for
vertices of
degree . A -edge matching in a graph is a set of independent edges.
In this thesis we prove the new result that a random
-irregular graph plus a random -edge matching is contiguous to a random
-regular graph, in the sense that
in the two spaces,
the same events have probability approaching 1 as .
This allows one to deduce properties, such as colourability,
of the random irregular graph from
the corresponding properties of the random regular one. The proof
applies the small subgraph conditioning method to the number of -edge matchings
in a random -regular graph.
The third problem is about the 3-colourability of
a random 5-regular graph. Call a colouring balanced
if the number of vertices of each colour
is equal, and locally rainbow if every vertex is adjacent to vertices
of all the other
colours. Using the small subgraph conditioning method, we give a
condition on the variance of the number of locally rainbow balanced 3-colourings which, if
satisfied, establishes that the chromatic number of the random 5-regular graph is
asymptotically almost surely equal to 3.
We also describe related work which provides evidence that the condition is
likely to be true.
The fourth problem is about the chromatic number of a random -regular
graph for fixed .
Achlioptas and Moore recently announced a proof that a random -regular
graph asymptotically almost surely has chromatic number , , or ,
where is the smallest integer satisfying . In
this thesis we prove that, asymptotically almost surely, it is not ,
provided a certain second moment condition holds.
The proof applies the small subgraph conditioning method to
the number of balanced -colourings, where a colouring is balanced
if the number of vertices of each colour is equal.
We also give evidence that suggests that the required
second moment condition is true
Properties of graphs with large girth
This thesis is devoted to the analysis of a class of
iterative probabilistic algorithms in regular graphs, called
locally greedy algorithms, which will provide bounds for
graph functions in regular graphs with large girth. This class is
useful because, by conveniently setting the parameters associated
with it, we may derive algorithms for some well-known graph
problems, such as algorithms to find a large independent set, a
large induced forest, or even a small dominating set in an input
graph G. The name ``locally greedy" comes from the fact that, in
an algorithm of this class, the probability associated with the
random selection of a vertex v is determined by the current
state of the vertices within some fixed distance of v.
Given r > 2 and an r-regular graph G, we determine the
expected performance of a locally greedy algorithm in G,
depending on the girth g of the input and on the degree r of
its vertices. When the girth of the graph is sufficiently large,
this analysis leads to new lower bounds on the independence number
of G and on the maximum number of vertices in an induced forest
in G, which, in both cases, improve the bounds previously known.
It also implies bounds on the same functions in graphs with large
girth and maximum degree r and in random regular graphs. As a
matter of fact, the asymptotic lower bounds on the cardinality of
a maximum induced forest in a random regular graph improve earlier
bounds, while, for independent sets, our bounds coincide with
asymptotic lower bounds first obtained by Wormald. Our result
provides an alternative proof of these bounds which avoids sharp
concentration arguments.
The main contribution of this work lies in the method presented
rather than in these particular new bounds. This method allows us,
in some sense, to directly analyse prioritised algorithms in
regular graphs, so that the class of locally greedy algorithms, or
slight modifications thereof, may be applied to a wider range of
problems in regular graphs with large girth