Some New 2-chromatic Steiner Quadruple Systems

A Steiner quadruple system of order v (briefly an SQS(v)) is an ordered pair (X,B) where X is a set of cardinality v whose elements are called points, and B is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. A k-proper coloring of a Steiner quadruple system (X, B) is a partition of the set X into k parts of color classes such that no block of B is contained in any color class. An SQS(v) is k-chromatic if it can be k-proper colored, but can not be (k-1)-proper colored. The unsolved problems on k-chromatic Steiner quadruple systems are 2-chromatic and 5-chromatic. In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic SQS(v) for all v≡4 or 8 (mod 12). In this thesis, we construct a few 2-chromatic candelabra quadruple systems by using some special automorphism groups. With the aid of a recursive construction for 2-chromatic SQSs, we proved that a 2-chromatic SQS(v) exists if v≡4 or 8 (mod 12), or if v≡2 or 10 (mod 24), or v=22.