Title:

Baranyai partitions and Kneser graphs

The Kneser graph K(n,r) has vortex set consisting of all the rsets of a fixed nset with two vertices being adjacent if the corresponding sets are disjoint. This thesis is mainly concerned with structural properties of this intriguing family of graphs and related combinatorial objects. As well as the general Kneser graph we study the Odd graphs, that is the subfamily of the Kneser graphs formed by taking n = 2r + 1, and subgraphs of the discrete cube related to K(n,r) via graph homomorphisms. In particular, we will be concerned with the existence of Hamilton cycles, and related issues such as cycle lengths and 2factors in these graphs. Among the results which we prove in this area are the existence of cycles through a proportion arbitrarily close to 1 of the vertices of the Middle Two Layers graph and the Kneser graph, the existence of a 2factorisation of the Odd graph, a second proof (following Chen) of the existence of a Hamilton cycle in K(n,r) when n ³ 3r, and an inductive construction which extends the range of values for which the Kneser graph is known to be Hamiltonian. We also study partitions of the vertex set of K(3k,3), where k is an integer, into complete graphs of order k. The existence of such a partitioning is a consequence of Baranyai’s Partition theorem. We address the question of whether such a partitioning can be made to satisfy a certain natural ‘separatedness’ condition, as conjectured by FonderFlaass. Our main result here is that FonderFlaass’ conjecture fails for n = 12.
