Partitions of combinatorial structures

• Main Author: Patel, Viresh
• Published: London School of Economics and Political Science (University of London) 2009
• Subjects: 512.7
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.645899

In this thesis we explore extremal, structural, and algorithmic problems involving the partitioning of combinatorial structures. We begin by considering problems from the theory of graph cuts. It is well known that every graph has a cut containing at least half its edges. We conjecture that (except for one example), given any two graphs on the same vertex set, we can partition the vertices so that at least half the edges of each graph go across the partition. We give a simple algorithm that comes close to proving this conjecture. We also prove, using probabilistic methods, that the conjecture holds for certain classes of graphs. We consider an analogue of the graph cut problem for posets and determine which graph cut results carry over to posets. We consider both extremal and algorithmic questions, and in particular, we show that the analogous maxcut problem for posets is polynomial-time solvable in contrast to the maxcut problem for graphs, which is NP-complete. Another partitioning problem we consider is that of obtaining a regular partition (in the sense of the Szemeredi Regularity Lemma) for posets, where the partition respects the order of the poset. We prove the existence of such order-preserving, regular partitions for both the comparability graph and the covering graph of a poset, and go on to derive further properties of such partitions. We give a new proof of an old result of Frankl and Furedi, which characterises all 3-uniform hypergraphs for which every set of 4 vertices spans exactly 0 or 2 edges. We use our new proof to derive a corresponding stability result. We also look at questions concerning an analogue of the graph linear extension problem for posets.