# The strong perfect graph theorem

@article{Murray2019TheSP, title={The strong perfect graph theorem}, author={A. M. Murray}, journal={100 Years of Math Milestones}, year={2019} }

In 1960 Berge came up with the concept of perfect graphs, and in doing so, conjectured some characteristics about them. 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 [2]. Two conjectures are now known as the Perfect Graph Theorem and the Strong Perfect Graph Theorem. Both of these theorems make detemining if a graph is perfect much easier than using the standard definition. Simply looking at any graph…

## 235 Citations

### Improved Algorithms for Recognizing Perfect Graphs and Finding Shortest Odd and Even Holes

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It is proved that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph.

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- MathematicsCOCOON
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This paper shows that the chromatic number of a (P 6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G, and shows that there is exactly one 6-vertex-critical (P6, diamond, K6)- free graph.

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- 2018

The Gyárfás-Sumner conjecture [29, 42] states that for every tree T , there is a function f such that for every graph G with no induced subgraph isomorphic to T , the chromatic number of G is at most…

### M2 Internship Report Structure of Even-Hole-Free Graphs

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An even-hole-free graph is a graph that has no induced even cycles of length at least 4. This class is structurally similar to the class of perfect graphs, which was one initial motivation for their…

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Let P n and K n denote the induced path and complete graph on n vertices, respectively. The kite is the graph obtained from a P 4 by adding a vertex and making it adjacent to all vertices in the P 4…

### Submodular functions and perfect graphs

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- 2021

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced…