Progress on perfect graphs

Abstract. A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle.

[1]  Aaron D. Wyner,et al.  The Zero Error Capacity of a Noisy Channel , 1993 .

[2]  Gérard Cornuéjols,et al.  Decomposing Berge Graphs Containing Proper Wheels , 2002 .

[3]  Robin Thomas,et al.  Tutte's Edge-Colouring Conjecture , 1997, J. Comb. Theory, Ser. B.

[4]  Colin McDiarmid,et al.  Graph Imperfection II , 2001, J. Comb. Theory, Ser. B.

[5]  Vasek Chvátal,et al.  Bull-free Berge graphs are perfect , 1987, Graphs Comb..

[6]  László Lovász,et al.  Matching structure and the matching lattice , 1987, J. Comb. Theory, Ser. B.

[7]  Alan Tucker,et al.  The strong perfect graph conjecture and an application to a municipal routing problem , 1972 .

[8]  D. R. Fulkerson,et al.  Anti-blocking polyhedra , 1972 .

[9]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[10]  D. R. Fulkerson,et al.  Blocking and anti-blocking pairs of polyhedra , 1971, Math. Program..

[11]  Robin Thomas,et al.  PERMANENTS, PFAFFIAN ORIENTATIONS, AND EVEN DIRECTED CIRCUITS , 1997, STOC 1997.

[12]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[13]  László Lovász,et al.  Entropy splitting for antiblocking corners and perfect graphs , 1990, Comb..

[14]  William McCuaig,et al.  Pólya's Permanent Problem , 2004, Electron. J. Comb..

[15]  A. Tucker,et al.  Perfect Graphs and an Application to Optimizing Municipal Services , 1973 .

[16]  G. Tinhofer.  Graphen und Matrizen , 1976 .

[17]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[18]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[19]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[20]  D. König.  Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre , 1916 .

[21]  K. Wagner.  Über eine Eigenschaft der ebenen Komplexe , 1937 .

[22]  Jeff Kahn,et al.  Entropy and sorting , 1992, STOC '92.

[23]  Bruce A. Reed,et al.  Channel assignment and weighted coloring , 2000, Networks.

[24]  Colin McDiarmid,et al.  Graph Imperfection , 2001, J. Comb. Theory, Ser. B.

[25]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[26]  Vasek Chvátal,et al.  Star-cutsets and perfect graphs , 1985, J. Comb. Theory, Ser. B.

[27]  Florian Roussel,et al.  About Skew Partitions in Minimal Imperfect Graphs , 2001, J. Comb. Theory, Ser. B.

[28]  Gérard Cornuéjols.  The Strong Perfect Graph Conjecture , 2002 .

[29]  Gérard Cornuéjols,et al.  Compositions for perfect graphs , 1985, Discret. Math..

[30]  Manfred W. Padberg,et al.  Perfect zero–one matrices , 1974, Math. Program..

[31]  N. Biggs.  MATCHING THEORY (Annals of Discrete Mathematics 29) , 1988 .

[32]  Chính T. Hoàng,et al.  Some properties of minimal imperfect graphs , 1996, Discret. Math..

[33]  S. E. Markosyan,et al.  ω-Perfect graphs , 1990 .

[34]  L. Lovász.  Normal Hypergraphs and the Weak Perfect Graph Conjecture , 1984 .

[35]  Gérard Cornuéjols,et al.  Graphs without odd holes, parachutes or proper wheels: a generalization of Meyniel graphs and of line graphs of bipartite graphs , 2003, J. Comb. Theory, Ser. B.

[36]  Bolian Liu,et al.  Combinatorial Properties of Matrices , 2000 .