This thesis deals with random graph processes. More precisely it deals with two
random graph processes which create H free graphs. The first of these processes
is the random Helimination process which starts from the complete graph and in
every step removes an edge uniformly at random from the set of edges which are
found in a copy of H. The second is the Hfree random graph process which starts
from the empty graph and in every step an edge chosen uniformly at random from
the set of edges which when added to the graph would not create a copy of H is
inserted.
We consider these graph processes for several classes of graphs H, for example
strictly two balanced graphs. The class of strictly two balanced graphs includes
among others cycles and complete graphs.
We analysed the Helimination process, when H is strictly 2balanced. For this
class we show the typical number of edges found at the end of the process. We
also consider the sub graphs created by the process and its independence number.
We also managed to show the expected number of edges in the H elimination pro
cess when H = Ki, the graph created from the complete graph on 4 vertices by
removing an edge and when H = K34 where K34 is created from the complete bi
partite graph with 3 vertices in one partition and'4 vertices in the second partition,
by removing an edge.
In case of the H free process we considered the case when H is the triangle and
showed that the trianglefree random graph process only creates sparse subgraphs.
Finally we have improved the lower bound on the length of the K34free random
graph process. '
