In graph theory, the **strong perfect graph theorem** is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither odd holes (odd-length induced cycles) nor odd antiholes (complements of odd holes). It was conjectured by Claude Berge in 1961. A proof by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas was announced in 2002^{ [1] } and published by them in 2006.

The proof of the strong perfect graph theorem won for its authors a $10,000 prize offered by Gérard Cornuéjols of Carnegie Mellon University^{ [2] } and the 2009 Fulkerson Prize.^{ [3] }

A perfect graph is a graph in which, for every induced subgraph, the size of the maximum clique equals the minimum number of colors in a coloring of the graph; perfect graphs include many well-known graph classes including the bipartite graphs, chordal graphs, and comparability graphs. In his 1961 and 1963 works defining for the first time this class of graphs, Claude Berge observed that it is impossible for a perfect graph to contain an odd hole, an induced subgraph in the form of an odd-length cycle graph of length five or more, because odd holes have clique number two and chromatic number three. Similarly, he observed that perfect graphs cannot contain odd antiholes, induced subgraphs complementary to odd holes: an odd antihole with 2*k* + 1 vertices has clique number *k* and chromatic number *k* + 1, which is again impossible for perfect graphs. The graphs having neither odd holes nor odd antiholes became known as the Berge graphs.

Berge conjectured that every Berge graph is perfect, or equivalently that the perfect graphs and the Berge graphs define the same class of graphs. This became known as the strong perfect graph conjecture, until its proof in 2002, when it was renamed the strong perfect graph theorem.

Another conjecture of Berge, proved in 1972 by László Lovász, is that the complement of every perfect graph is also perfect. This became known as the perfect graph theorem, or (to distinguish it from the strong perfect graph conjecture/theorem) the weak perfect graph theorem. Because Berge's forbidden graph characterization is self-complementary, the weak perfect graph theorem follows immediately from the strong perfect graph theorem.

The proof of the strong perfect graph theorem by Chudnovsky et al. follows an outline conjectured in 2001 by Conforti, Cornuéjols, Robertson, Seymour, and Thomas, according to which every Berge graph either forms one of five types of basic building block (special classes of perfect graphs) or it has one of four different types of structural decomposition into simpler graphs. A minimally imperfect Berge graph cannot have any of these decompositions, from which it follows that no counterexample to the theorem can exist.^{ [4] } This idea was based on previous conjectured structural decompositions of similar type that would have implied the strong perfect graph conjecture but turned out to be false.^{ [5] }

The five basic classes of perfect graphs that form the base case of this structural decomposition are the bipartite graphs, line graphs of bipartite graphs, complementary graphs of bipartite graphs, complements of line graphs of bipartite graphs, and double split graphs. It is easy to see that bipartite graphs are perfect: in any nontrivial induced subgraph, the clique number and chromatic number are both two and therefore both equal. The perfection of complements of bipartite graphs, and of complements of line graphs of bipartite graphs, are both equivalent to Kőnig's theorem relating the sizes of maximum matchings, maximum independent sets, and minimum vertex covers in bipartite graphs. The perfection of line graphs of bipartite graphs can be stated equivalently as the fact that bipartite graphs have chromatic index equal to their maximum degree, proven by Kőnig (1916). Thus, all four of these basic classes are perfect. The double split graphs are a relative of the split graphs that can also be shown to be perfect.^{ [6] }

The four types of decompositions considered in this proof are 2-joins, complements of 2-joins, balanced skew partitions, and homogeneous pairs.

A 2-join is a partition of the vertices of a graph into two subsets, with the property that the edges spanning the cut between these two subsets form two vertex-disjoint complete bipartite graphs. When a graph has a 2-join, it may be decomposed into induced subgraphs called "blocks", by replacing one of the two subsets of vertices by a shortest path within that subset that connects one of the two complete bipartite graphs to the other; when no such path exists, the block is formed instead by replacing one of the two subsets of vertices by two vertices, one for each complete bipartite subgraph. A 2-join is perfect if and only if its two blocks are both perfect. Therefore, if a minimally imperfect graph has a 2-join, it must equal one of its blocks, from which it follows that it must be an odd cycle and not Berge. For the same reason, a minimally imperfect graph whose complement has a 2-join cannot be Berge.^{ [7] }

A skew partition is a partition of a graph's vertices into two subsets, one of which induces a disconnected subgraph and the other of which has a disconnected complement; Chvátal (1985) had conjectured that no minimal counterexample to the strong perfect graph conjecture could have a skew partition. Chudnovsky et al. introduced some technical constraints on skew partitions, and were able to show that Chvátal's conjecture is true for the resulting "balanced skew partitions". The full conjecture is a corollary of the strong perfect graph theorem.^{ [8] }

A homogeneous pair is related to a modular decomposition of a graph. It is a partition of the graph into three subsets *V*_{1}, *V*_{2}, and *V*_{3} such that *V*_{1} and *V*_{2} together contain at least three vertices, *V*_{3} contains at least two vertices, and for each vertex *v* in *V*_{3} and each *i* in {1,2} either *v* is adjacent to all vertices in *V*_{i} or to none of them. It is not possible for a minimally imperfect graph to have a homogeneous pair.^{ [9] } Subsequent to the proof of the strong perfect graph conjecture, Chudnovsky (2006) simplified it by showing that homogeneous pairs could be eliminated from the set of decompositions used in the proof.

The proof that every Berge graph falls into one of the five basic classes or has one of the four types of decomposition follows a case analysis, according to whether certain configurations exist within the graph: a "stretcher", a subgraph that can be decomposed into three induced paths subject to certain additional constraints, the complement of a stretcher, and a "proper wheel", a configuration related to a wheel graph, consisting of an induced cycle together with a hub vertex adjacent to at least three cycle vertices and obeying several additional constraints. For each possible choice of whether a stretcher or its complement or a proper wheel exists within the given Berge graph, the graph can be shown to be in one of the basic classes or to be decomposable.^{ [10] } This case analysis completes the proof.

- ↑ Mackenzie (2002); Cornuéjols (2002).
- ↑ Mackenzie (2002).
- ↑ "2009 Fulkerson Prizes" (PDF),
*Notices of the American Mathematical Society*: 1475–1476, December 2011. - ↑ Cornuéjols (2002), Conjecture 5.1.
- ↑ Reed (1986); Hougardy (1991); Rusu (1997); Roussel, Rusu & Thuillier (2009), section 4.6 "The first conjectures".
- ↑ Roussel, Rusu & Thuillier (2009), Definition 4.39.
- ↑ Cornuéjols & Cunningham (1985); Cornuéjols (2002), Theorem 3.2 and Corollary 3.3.
- ↑ Seymour (2006); Roussel, Rusu & Thuillier (2009), section 4.7 "The skew partition"; Cornuéjols (2002), Theorems 4.1 and 4.2.
- ↑ Chvátal & Sbihi (1987); Cornuéjols (2002), Theorem 4.10.
- ↑ Cornuéjols (2002), Theorems 5.4, 5.5, and 5.6; Roussel, Rusu & Thuillier (2009), Theorem 4.42.

In the mathematical field of graph theory, a **bipartite graph** is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the *parts* of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

This is a **glossary of graph theory terms**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

In graph theory, 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. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if for all we have .

In graph theory, the **perfect graph theorem** of László Lovász states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by Berge, and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs.

In the mathematical discipline of graph theory, the **line graph** of an undirected graph *G* is another graph *L*(*G*) that represents the adjacencies between edges of *G*. The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) used the construction before this. Other terms used for the line graph include the **covering graph**, the **derivative**, the **edge-to-vertex dual**, the **conjugate**, the **representative graph**, and the **ϑ-obrazom**, as well as the **edge graph**, the **interchange graph**, the **adjoint graph**, and the **derived graph**.

In graph theory, an **edge coloring** of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The **edge-coloring problem** asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the **chromatic index** of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

In graph theory, a **Meyniel graph** is a graph in which every odd cycle of length five or more has at least two chords, edges connecting non-consecutive vertices of the cycle. The chords may be uncrossed or they may cross each other, as long as there are at least two of them.

In mathematics, in the areas of order theory and combinatorics, **Dilworth's theorem** characterizes the **width** of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician Robert P. Dilworth (1950).

**Paul Seymour** is currently a professor at Princeton University; half in the department of mathematics and half in the program in applied and computational math. His research interest is in discrete mathematics, especially graph theory. He was responsible for progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, and claw-free graphs. Many of his recent papers are available from his website.

In graph theory, an **induced subgraph** of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset.

In graph theory, a **rook's graph** is a graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and each edge represents a legal move from one square to another. The same graphs can be defined mathematically as the Cartesian products of two complete graphs or as the line graphs of complete bipartite graphs.

In graph theory, a mathematical discipline, a **factor-critical graph** is a graph with n vertices in which every subgraph of *n* − 1 vertices has a perfect matching.

In graph theory, a branch of mathematics, a **split graph** is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by Földes and Hammer, and independently introduced by Tyshkevich and Chernyak (1979).

In graph theory, an area of mathematics, a **claw-free graph** is a graph that does not have a claw as an induced subgraph.

In graph theory, a branch of discrete mathematics, a **distance-hereditary graph** is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.

**Maria Chudnovsky** is an Israeli-American mathematician working on graph theory and combinatorial optimization. She is a 2012 MacArthur Fellow.

In the mathematical field of graph theory, the **bull graph** is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges.

In graph theory, a branch of mathematics, the **Herschel graph** is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.

In graph theory, a **skew partition** of a graph is a partition of its vertices into two subsets, such that the induced subgraph formed by one of the two subsets is disconnected and the induced subgraph formed by the other subset is the complement of a disconnected graph. Skew partitions play an important role in the theory of perfect graphs.

In the mathematical discipline of graph theory, **Petersen's theorem**, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows:

Petersen's Theorem.Every cubic, bridgeless graph contains a perfect matching.

- Berge, Claude (1961), "Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind",
*Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe*,**10**: 114. - Berge, Claude (1963), "Perfect graphs",
*Six Papers on Graph Theory*, Calcutta: Indian Statistical Institute, pp. 1–21. - Chudnovsky, Maria (2006), "Berge trigraphs",
*Journal of Graph Theory*,**53**(1): 1–55, doi:10.1002/jgt.20165, MR 2245543 . - Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), "The strong perfect graph theorem",
*Annals of Mathematics*,**164**(1): 51–229, arXiv: math/0212070 , doi:10.4007/annals.2006.164.51, MR 2233847 . - Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2003), "Progress on perfect graphs",
*Mathematical Programming*, Series B.,**97**(1–2): 405–422, CiteSeerX 10.1.1.137.3013 , doi:10.1007/s10107-003-0449-8, MR 2004404 . - Chvátal, Václav (1985), "Star-cutsets and perfect graphs",
*Journal of Combinatorial Theory*, Series B,**39**(3): 189–199, doi: 10.1016/0095-8956(85)90049-8 , MR 0815391 . - Chvátal, Václav; Sbihi, Najiba (1987), "Bull-free Berge graphs are perfect",
*Graphs and Combinatorics*,**3**(2): 127–139, doi:10.1007/BF01788536, MR 0932129 . - Cornuéjols, Gérard (2002), "The strong perfect graph conjecture",
*Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002)*(PDF), Beijing: Higher Ed. Press, pp. 547–559, MR 1957560 . - Cornuéjols, G.; Cunningham, W. H. (1985), "Compositions for perfect graphs",
*Discrete Mathematics*,**55**(3): 245–254, doi:10.1016/S0012-365X(85)80001-7, MR 0802663 . - Hougardy, S. (1991),
*Counterexamples to three conjectures concerning perfect graphs*, Technical Report RR870-M, Grenoble, France: Laboratoire Artemis-IMAG, Universitá Joseph Fourier. As cited by Roussel, Rusu & Thuillier (2009). - Kőnig, Dénes (1916), "Gráfok és alkalmazásuk a determinánsok és a halmazok elméletére",
*Matematikai és Természettudományi Értesítő*,**34**: 104–119. - Lovász, László (1972a), "Normal hypergraphs and the perfect graph conjecture",
*Discrete Mathematics*,**2**(3): 253–267, doi:10.1016/0012-365X(72)90006-4 . - Lovász, László (1972b), "A characterization of perfect graphs",
*Journal of Combinatorial Theory*, Series B,**13**(2): 95–98, doi:10.1016/0095-8956(72)90045-7 . - Mackenzie, Dana (July 5, 2002), "Mathematics: Graph theory uncovers the roots of perfection",
*Science*,**297**(5578): 38, doi:10.1126/science.297.5578.38, PMID 12098683 . - Reed, B. A. (1986),
*A semi-strong perfect graph theorem*, Ph.D. thesis, Montréal, Québec, Canada: Department of Computer Science, McGill University. As cited by Roussel, Rusu & Thuillier (2009). - Roussel, F.; Rusu, I.; Thuillier, H. (2009), "The strong perfect graph conjecture: 40 years of attempts, and its resolution",
*Discrete Mathematics*,**309**(20): 6092–6113, doi:10.1016/j.disc.2009.05.024, MR 2552645 . - Rusu, Irena (1997), "Building counterexamples",
*Discrete Mathematics*,**171**(1–3): 213–227, doi:10.1016/S0012-365X(96)00081-7, MR 1454452 . - Seymour, Paul (2006), "How the proof of the strong perfect graph conjecture was found" (PDF),
*Gazette des Mathématiciens*(109): 69–83, MR 2245898 .

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.