Presentation on theme: "Ramsey Theory and Applications CS 594 Graph Theory Presented by: Kai Wang."— Presentation transcript:
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
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
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?