On set intersection representations of graphs

Abstract

The intersection dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets A x⊆{1,...,t} of labels to vertices × so that any two vertices x and y from different parts are adjacent if and only if \A x∩Ay\eL. The weight of such a representation is the sum σX\AX\ over all vertices x. We exhibit explicit bipartite nxn graphs whose intersection dimension is (i) at least n 1/|/-! with respect to any type L, (ii) at least «n with respect to any type of the form L = {k,k+'\,...}, and (iii) at least n 1/|R| with respect to any type of the form L = {k\kmo6p∈H}, where p is a prime number. We also show that any intersection representation of a Hadamard graph must have weight about nInn/In Inn, independent on the used type L. Finally, we formulate several problems about intersection dimensions of graphs related to some basic open problems in the complexity of boolean functions. © 2009 wiley Periodicals, © 2009 Wiley Periodicals, Inc.