On minimal imperfect graphs with circular symmetry

Abstract

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 Sebö. 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. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998