Ramsey Theory and Applications CS 594 Graph Theory Presented by: Kai Wang

Backgrounds and Definitions

From Pigeonhole Principle to Ramsey Theory

What is this all about?  Part of extremal combinatorics: smallest configuration with special structural properties.  In extremal graph theory, many problems are difficult.

Applications of Pigeonhole Principle

History

Historical Perspectives  Ramsey theory was initially studied in the context of propositional logic (1928)  Theodore S. Motzkin: “Complete disorder is impossible”.  Become known after Paul Erdos and George Szekeres (1935) applied it in graph theory

Some Notations

Ramsey Theorem

Ramsey Number Examples  R(3,3;2)=6  R(3,3,3;2)=17  R(4,4;3)=13  We don’t know the exact value of R(5,5;2), although it is between 43 and 49.  Little is known about how to calculate the exact value of Ramsey numbers, because currently we only know of the brute-force method, which is impractical.

Graph Ramsey Theory

Graph Ramsey Number Examples

Applications  Potential in areas that can apply pigeon hole principles  Graph Ramsey theory adds many natural examples to computational complexity classes

A decision problem

Another decision problem

Application in information retrieval problem

Part II  Edmonds’ maximum matching algorithm

Maximum Matching Problem

Berge’s Lemma  An augmenting path in a graph G with respect to a matching M is an alternating path with the two endpoints exposed (unmatched).  A matching M in G is of maximum cardinality if and only if (G,M) does not contain an augmenting path.

Key Concepts: Blossom (courtesy of

Blossom Algorithm  If the contracted graph is G’, then a maximum matching of G’ corresponds to a maximum matching of G.  We first get a maximum matching M’ of G’, then by expanding M’ we get a maximum matching M of G.  Complexity: O(n^4)

Other Implementations  One of the famous is Gabow’s labeling implementation of Edmonds’ algorithm by avoiding expanding M’ in the contracted graph G’.  Complexity: O(n^3)

References  Douglas B. West: Introduction to Graph Theory, Section 8.3  Vera Rosta: Ramsey Theory Applications, The Electronic Journal of Combinatorics, 2004  Edmonds, Jack (1965). "Paths, trees, and flowers". Canad. J. Math. 17: 449–467  Harold N. Gabow: An Efficient implementation of Edmonds' Algorithm for Maximum Matching on Graphs, Journal of the ACM, Volume 23 Issue 2, 1976

Homework  Prove or disprove: R(p,2;2)=p when p>=2.  Prove or disprove: R(4,3;2)>=10.  What is the relationship between maximum matching size and minimum vertex cover size?

Thank You  Questions?