Partitionable graphs arising from near-factorizations of finite groups Original Research Article

In 1979, two constructions for making partitionable graphs were introduced in (by Chvátal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A,B) of a finite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to be a Cayley graph of the cyclic group Zn, with connection set (A−A)⧹{0}, for a near-factorization (A,B) of Zn. Since a counter-example to the Strong Perfect Graph Conjecture would be a partitionable graph (Padberg, Math. Programming 6 (1974) 180), any ‘new’ construction for making partitionable graphs is of interest. In this paper, we investigate the near-factorizations of finite groups in general, and their associated Cayley graphs which are all partitionable. In particular, we show that near-factorizations of the dihedral groups produce every CGPW graph of even order. We present some results about near-factorizations of finite groups which imply that a finite abelian group with a near-factorization (A,B) such that |A|⩽4 must be cyclic (already proved by De Caen et al. (Ars Combin. 29 (1990) 53)). One of these results may be used to speed up exhaustive calculations. At last, we prove that there is no counter-example to the Strong Perfect Graph Conjecture arising from near-factorizations of a finite abelian group of even order.