E xtremal S ubgraphs of R andom G raphs. Konstantinos Panagiotou Institute of Theoretical Computer Science ETH Zürich. (joint work with Graham Brightwell and Angelika Steger ). Maximum bipartite subgraph. Motivation. Maximum triangle-free subgraph. Maximum bipartite subgraph.
Institute ofTheoretical Computer Science
(joint work with Graham Brightwell and Angelika Steger)
set of maximum triangle-free subgraphs
set of maximum bipartite subgraphs
size of a maximum triangle-free subgraph
size of a maximum bipartite subgraph
here: [Mantel 1907]
we know from random graph theory:
is aas. a forest
no edge within the larger class can be added (without creating a triangle)
Pr[edge can be added]
Pr[no edge can be added]
Erdös, Kleitman, Rothschild (1976)
Almost all large triangle-freegraphs are bipartite.
Prömel, Steger (1996), Steger (2005),
Osthus, Prömel, Taraz (2003):
Could a similar picture be true for the random graph ?
Babai, Simonovits, Spencer (1990)
Question (Babai, Simonovits, Spencer):
Do we have
for all constant ? What if ?
Lemma (sparse case)
Pair is regular
Vertices which do
not fulfill the
“Too many” missing
• All „exceptional sets“ are small
• „Remainder“ is very sparse
How does a maximum triangle-free subgraph - of look like?
is a bipartition
Observe: has to contain all edges of
between except possibly some edges
has to be a partition that is within to
an optimum partition
Howlikelyisitthatonecanadd a singlerededge after removing
atmostedgesfrom (withoutcreating a triangle)?
disjoint triangles with and
Need: number of partitions to be considered
Two points of view
Properties of random graphs (for ):
(next three slides)
Conjecture (Erdös 1976):
Everytriangle-free graph on n vertices can be made bipartite by deleting at most n2/25 edges.
If true, then best possible.