# 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…

## 243 Citations

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

- Computer ScienceArXiv
- 2022

Improved algorithms for detecting or ﬁnding induced subgraphs in G, a graph that can be obtained by deleting a set of vertices together with its incident edges, are shown.

Colouring perfect graphs with bounded clique number

- MathematicsJ. Comb. Theory, Ser. B
- 2017

χ‐bounded families of oriented graphs

- MathematicsJ. Graph Theory
- 2018

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.

Colouring graphs with no induced six-vertex path or diamond

- MathematicsCOCOON
- 2021

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.

Clique-perfectness and balancedness of some graph classes

- MathematicsInt. J. Comput. Math.
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This work gives linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness of P4-tidy graphs and alinear-time algorithm for computing a maximumClique-independent set and a minimum cliques-transversal set for any P4 -tidy graph.

M2 Internship Report Structure of Even-Hole-Free Graphs

- Mathematics
- 2018

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…

Coloring ($P_5$, kite)-free graphs

- Mathematics
- 2022

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

- Mathematics
- 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…

On graphs with no induced bull and no induced diamond

- Mathematics
- 2021

A bull is the graph obtained by adding two pendant edges at different vertices of a triangle. A diamond is the graph obtained from a K4 by deleting an edge. In this paper, we study the upper bound…

Simple Proofs of the Strong Perfect Graph Theorem Using Polyhedral Approaches and Proving P=NP as a Conclusion

- Mathematics2020 International Conference on Computational Science and Computational Intelligence (CSCI)
- 2020

The strong perfect graph theorem is the proof of the famous Berge’s conjecture that the graph is perfect if and only if it is free of odd holes and odd anti-holes. The conjecture was settled after 40…