The Uniformity Space of Hypergraphs

Abstract

For a hypergraph H=(V,E) and a field F, a weighting of H is a map f:V ?F. A weighting is called stable if there is some k ? F such that the sum of the weights on each edge of H is equal to k. The set of all stable weightings of H forms a vector space over F. This vector space is termed the uniformity space of H over F, denoted U(H,F), and its dimension is called the uniformity dimension of H over F. This thesis is concerned with several problems relating to the uniformity space of hypergraphs. For several families of hypergraphs, simple ways of computing their uniformity dimension are found. Also, the uniformity dimension of random l-uniform hypergraphs is investigated. The stable weightings of the spanning trees of a graph are determined, and lastly, a notion of critical uniformity dimension is introduced and explored.