As with many results discovered through structural methods, the theorem s proof is long and technical. In this paper, we present a formal framework for working with finite simple graphs. It is easy to check that none of the forbidden induced subgraphs in an innocent graph has a strong stable set, and so they are not strongly perfect. A partitionable graph without a small transversal has been constructed by by chvatal, graham, perold, and whitesides op. The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by lovasz. The strong perfect graph theorem states that the converse is also true. Their proof, which was published in the annals of mathematics in 2006, filled 150 pages. This continues from our planar graphs video with maria chudnovsky at s. The strong perfect graph theorem of chudnovsky et al. Polyhedral results on the stable set problem in graphs containing even or odd pairs. Specifically, i have expertise with a measure called the treewidth of a graph. The strong perfect graph conjecture berge, 1961 asserts that a graph is perfect if and only if. In 1961, berge proposed two excellent conjectures about perfect graphs.
Because this characterization is unaffected by graph complementation, it immediately implies the weak perfect graph theorem. The criterion we used to include a class of perfect graphsin this survey is that its studybe motivated by making progress towards a proof of the strong perfect graph conjecture. The theory of perfect graphs relates the concept of graph colorings to the concept of cliques. Introduction the theory of perfect graphs relates the concept of graph colorings to the concept of cliques. A re nement of this theorem gives an algorithm to decide whether gis perfect. Along with his collaborators, they were able to prove that a graph is perfect if and only if the graph, g, has no induced subgraph of g that is an odd cycle of length at least five or the. 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 oddlength induced cycles nor. The original proof was 150 pages long it has since been shortened somewhat 3. By the strong perfect graph theorem, we know that a minimal imperfect graph has no odd pair of maximal cliques. Then gis strongly perfect if and only if gis innocent. If g is a minimum imperfect graph, then g admits no balanced skew partition. In this paper, we introduce the concept of a perfect graph as well as a number of graph classes that are always perfect.
The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by lov\asz. May 18, 2016 maria has the most famous phd dissertation in the recent history of graph theory. Graphs g and h with the same set of vertices are called p,isomorphic if the following condition holds. For certain classes of graphs, including k 1,3free graphs, it is straightforward to validate this criterion and thus establish the strong perfect graph theorem for such graphs. A short proof of the wonderful lemma new jersey research. We next introduce both theweak perfect graph theorem and the strong perfect graph theorem and provide a proof of the weak perfect graph theorem.
Our proof of theorem 1 relies on the following result which is of independent interest. The strong perfect graph conjecture berge, 1961 asserts that a. There is a part of graph theory which actually deals with graphical drawing and presentation of graphs, brie. A key tool that is used many many times throughout the proof is what the authors of 2 lovingly call the wonderful lemma, that was first proved in 5 it was also proved independently by the authors of 2 in joint work. Their proof also leads to an efficient algorithm for finding 4colorings of planar graphs. However, the strong perfect graph theorem allows us to look at one single characteristic, the graph s holes, in order to determine if the graph is perfect 1. Paul seymour mathematician interesting stories about. Chudnovsky, robertson, seymour, thomas 2002 gis perfect i it has no odd hole and no odd antihole of size at least 5.
A graph g is perfect if for every induced subgraph h, the chromatic number of h equals the size of the largest complete subgraph of h, and g is berge if no induced subgraph of g is an odd cycle of length at least five or the complement of one. A hole said to be odd if it contains an odd number of. Strong perfect graph theorem an undirected graph is perfect if and only if it does not contain an induced. A short proof of the wonderful lemma chudnovsky 2018. The proof of the strong perfect graph theorem by chudnovsky et al. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem. What are the most overpowered theorems in mathematics. This result had been conjectured by berge 1961, 1963, and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem. The strong perfect graph conjecture spgc states that a graph is perfect if and only if it does not contain an odd hole or its complement.
Chudnovsky, robertson, seymour, thomas 2002 a graph is perfect if and only if it is a berge graph. Strong and weak perfect graph conjectures, respectively. The strong perfect graph conjecture spgc was certainly one of the most challenging conjectures in graph theory. The proof of the converse statement is very long and relies on structural graph theory. If g and h are p4isomorphic graphs and d is an invariant disc in v, then for all m in mthere exists some element d in d such that and is an invariant pair. Perfect graphs chapter 7 topics in chromatic graph theory. The strong perfect graph theorem for a class of partitionable. The strong perfect graph theorem annals of mathematics. The strong perfect graph theorem in may 2002, maria chudnovsky and paul seymour announced that they, building on earlier joint work with neil robertson and robin thomas, had completed the proof of the strong perfect graph conjecture. The strong perfect graph theorem 55 were used by roussel and rubio in their proof 23 of 2. Lovasz gave in 1971 a proof of the wpgc 65, but the spgc remained. An induced odd circuit is an oddlength, circular sequence of edges having no shortcircuit edges across it, while g.
The first of these three approaches yielded the first and to date only proof of the spgc. The strong perfect graph conjecture for panfree graphs. The strong perfect graph theorem by maria chudnovsky, neil robertson, paul seymour, robin thomas annals of mathematics, 2006 a graph g is perfect if for every induced subgraph h, the chromatic number of h equals the size of the largest complete subgraph of h, and g is berge if no induced subgraph of g is an odd cycle of length at least five. She helped prove the strong perfect graph theorem spgt, first posed by claude berge back in the 1960s. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The strong perfect graph theorem columbia university. Research on spgt and perfect graphs resulted in hundreds of papers and partial solutions before its resolution.
The strong perfect graph conjecture classes 2 and 3 are the same as class 4. Our primary application will be using perfect graphs to prove two order theory theorems. Aug 22, 2000 it follows that a proof of conjecture 3. The strong perfect graph conjecture berge, 1961 asserts that a graph is. The strong perfect graph conjecture berge, 1961 asserts that a graph is perfect if and only if it is berge. Introduction the theory of perfect graphs relates the concept of graph colorings to the. Theorem of the day the strong perfect graph theorem a graph g is perfect if and only if neither g nor its complement g. Perfect graph theorem and the strong perfect graph theorem and provides a proof of the weak theorem. How the proof of the strong perfect graph conjecture was found. The main result of this paper is a new shorter proof of theorem 2. In some special cases we prove this fact independently of the strong perfect graph. This led to a proof of berges strong perfect graph conjecture, and it was naturally expected to also lead to a polynomial time recognition algorithm for berge graphs, but that does not seem to be the case. This was known as the strong perfect graph conjecture and was finally answered in the affirmative in may, 2002.
A simple adjacency criterion is presented which, when satisfied, implies that a minimal imperfect graph is an odd hole or an odd antihole. The contributions to strong perfect graphs come from the proof of the strong perfect graph theorem proven in the paper the strong perfect graph theorem. A hole is an induced cycle of length at least four, and an antihole of a graph is a hole in the complement of a graph 3. If x sees all of gy, then y is a starcutset in g separating x from gxy. In graph theory, the perfect graph theorem of laszlo lovasz 1972a, 1972b states that an undirected graph is perfect if and only if its complement graph is also perfect. Using the strong perfect graph theorem we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Semi strong perfect graph theorem 229 observation 2. Perfect graphs structure and recognition department of.
The second, the strong perfect graph conjecture, received a great deal of attention over the past 40 years, but remained open until now, and is the main theorem of this paper. Efforts towards solving the problem have led to deep insights in the field of structural graph theory, where. Treewidth is at the heart of the proof of robertson and seymours deep graph structure theorem and has far reaching implications, contributing to proofs of the graph minor theorem and the strong perfect graph theorem, and a simplified proof of the four color. A constructive formalization of the weak perfect graph theorem. Using the strong perfect graph theorem 7 we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. The aim of this paper is to survey 120 classes of perfect graphs. Decomposing berge graphs containing proper wheels 2002. However, the strong perfect graph theorem allows us to look at one single characteristic, the graphs holes, in order to determine if the graph is perfect 1.
Graph theory uncovers the roots of perfection science. Oct 28, 2009 the strong perfect graph conjecture spgc was certainly one of the most challenging conjectures in graph theory. The weak perfect graph conjecture, later lovaszs theorem. This result is now known as the perfect graph theorem. An introduction to perfect graphs week 1 mathcamp 2011 last lecture, in our attempt to \classify the graphs of high chromatic number in a similar way to how we classi ed all of the bipartite graphs, we. The weak per fect graph conjectures was proved by lovasz 8 and is now known as the perfect graph theorem. We model finite simple graphs in the coq proof assistant by representing its vertices as a finite set over a countably infinite domain. The converse is the wellknown strong perfect graph conjecture. Transitiv orientierbare graphen frederic maffray and myriam preissmann. A stronger conjecture was made recently by conforti, cornuejols and vuskovic that every berge graph either falls into one of a few basic classes, or it has a kind of separation that cannot occur in a minimal imperfect graph. Polyhedral results on the stable set problem in graphs. The strong perfect graph conjecture holds true for pan free graphs. In 1960, berge formulated two conjectures about perfect graphs, one stronger than the other. A graph is strongly perfect if every induced subgraph has a strong stable set.
Recall that a hole is an induced subgraph that is a cycle, and an antihole is the complement of a hole. Abstract even and odd pairs are important tools in the study of perfect graphs and were instrumental in the proof of the strong perfect graph theorem. The weak perfect graph theorem wpgt states that a graph is perfect if and only if its complement is perfect. But for four graph theorists, it has a very specific value. It was in 2002 that chudnovsky along with seymour, then her ph. The weak perfect graph conjecture, later lov aszs theorem. The strong perfect graph theorem is the proof of the famous berges conjecture that the graph is perfect if and only if it is free of odd holes and odd antiholes. Nov 11, 2019 the strong perfect graph theorem paper.
Dec 04, 2002 a graph g is perfect if for every induced subgraph h, the chromatic number of h equals the size of the largest complete subgraph of h, and g is berge if no induced subgraph of g is an odd cycle of length at least 5 or the complement of one. A graph is perfect if and only if it does not contain an odd hole or an odd antihole. Maria has the most famous phd dissertation in the recent history of graph theory. In the spirit of kroneckers decomposition theorem for finite abelian groups 90. The theorem, originally conjectured by berge 1960, 1961, that a graph is perfect iff neither the graph nor its graph complement contains an odd graph cycle of. Treewidth is at the heart of the proof of robertson and seymours deep graph structure theorem and has far reaching implications, contributing to proofs of the graph minor theorem and the strong perfect graph theorem, and a simplified proof. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods.
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