subdivision of a big complete graph do not behave as well as
those without a minor of a big complete graph.
In fact, the graph minor theorem (a proof of Wagner's
conjecture) is not true if we replace the minor relation by the
subdivision relation. I.e, For every infinite sequence
G_1,G_2, ... of graphs, there exist distinct integers
i < j such that G_i is a minor of G_j, but if we replace
''minor" by ''subdivision", this is no longer true.
This is partially because we do not really know what the graphs
without a subdivision of a big complete graph look like.
In this talk, we shall discuss this issue. In particular,
assuming some moderate connectivity condition, we can say
something, which we will present in this talk.
Topics also include coloring graphs without a subdivision of a
large complete graph, and some algorithmic aspects. Some of the
results are joint work with Theo Muller.
deterministic statements using probabilistic tools. Two of the most famous
examples arise in number theory. these are: the first non-analytic proof
of the prime number theorem given by Erdos in the 1940s, and the recent
proof of the Hardy-Littlewood Conjecture (that there are arbitrarily long
arithmetic progressions of primes) by Green and Tao.
The method has also been succesfully applied in the field of graph
colouring. We survey some of the results thereby obtained.
The talk is targeted at a general audience. We will first define graph
colouring, explain the type of graph colouring problems which tend to
attract interest, and then explain the probabilistic tools which are
to solve them, and why we would expect the type of tools that are used to
be effective for solving the types of problems typically studied.