Strong Perfect Graph Theorem Ann Marie Murray Nebraska Wesleyan University May 4, 2016 Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 1 / 11 Clique and Induced Subgraph Definition Given any subset S of the vertices of a graph G(E; V ), the induced subgraph takes the vertex set S with the edges that connect these vertices in G.A clique is a subset of vertices of an undirected graph G(E; V ) such that every two distinct vertices are adjacent. Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 2 / 11 Chromatic Number Definition The chromatic number of any graph is the minimum number of colors needed to color the vertices of the graph such that no two adjacent vertices are colored the same. Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 3 / 11 Perfect Graph Definition (Berge, 1960) A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 4 / 11 Perfect Graph Theorem Theorem (Berge, 1960) The complement of a perfect graph is perfect. Lov´aszproved this theorem in 1972. Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 5 / 11 Definition A hole is a cycle in the graph, of length at least four, which is an induced subgraph. A antihole is a cycle in the complement of a graph, of length at least four, which is an induced subgraph. They are said to be odd if containing an odd number of vertices. Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 6 / 11 Strong Perfect Graph Theorem Theorem (Berge, 1961) A graph is perfect if and only if it contains no odd graph hole and no odd graph antihole Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 7 / 11 2001: Neil Robertson, Paul Seymour, Robin Thomas, Michele Conforti and G´erard Cornu´ejolsproposed an approach to consider five basic classes and four types of separations 2002: Maria Chudnovsky, Robertson, Seymour, and Thomas proved that if one of these classes or separations are found within the graph then it is perfect, and they also proved that every perfect graph must have one of the nine traits. Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 8 / 11 Bipartite Graph Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 9 / 11 Proof was not published until 2006 (45 years later) Recieved a 2009 Fulkerson Prize from the American Mathematical Society and a $10,000 prize from Cornu´ejolsat Carnegie Mellon University Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 10 / 11 Thanks Thank you for listening! References: 1 en.wikipedia.org/wiki/Strong_perfect_graph_theorem 2 integer.tepper.cmu.edu/webpub/optima.pdf 3 www.columbia.edu/~mc2775/perfect.pdf Ann Marie Murray (NWU) Strong Perfect Graph Theorem May 4, 2016 11 / 11.