Properties of graphs with large girth

Abstract

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.