Orthonormal Representations of H-Free Graphs

Abstract

Let x(1), ... , x(n) is an element of R-d be unit vectors such that among any three there is an orthogonal pair. Howlarge can n be as a function of d, and howlarge can the length of x(1)+ ... + x(n) be? The answers to these two celebrated questions, asked by Erdos and Lovasz, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovasz.-function and minimum semidefinite rank. In this paper, we study these parameters for general H-free graphs. In particular, we show that for certain bipartite graphs H, there is a connection between the Turan number of H and the maximum of v((G) over bar) over all H-free graphs G.ISSN:0179-5376ISSN:1432-044