On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge’s famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (”odd holes”) and their complements (”odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge’s Conjecture, as shown by Sebő. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called ”British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. ∗The second and third authors gratefully acknowledge the partial support by the Office of Naval Research (Grants N0001492F1375 and N0001492F4083) and by the Air Force Office of Scientific Research (Grant F49620-93-1-0041). The second author was also supported by NSF (grant INT 9321811) and NATO (grant CRG 931531). †Computer and Automation Institute, Hungarian Academy of Sciences, Kende u., Budapest, Hungary. E-mail: [email protected] ‡RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003, USA. E-mail: [email protected] §RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003, USA; On leave from the International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow, Russia. E-mail: [email protected] ¶CNRS, Laboratoire Leibniz, 46 avenue Félix Viallet, 38031 Grenoble Cedex, France. E-mail: [email protected] ‖CNRS, Laboratoire Leibniz, 46 avenue Félix Viallet, 38031 Grenoble Cedex, France. E-mail: [email protected]