New descriptions of the Lovász number, and the weak sandwich theorem

Abstract

In 1979, L. Lovász introduced the concept of an orthonormal representation of a graph, and also a related value, now popularly known as the Lovász number of the graph. One of the remarkable properties of the Lovász number is that it lies sandwiched between the stability number of the graph and the chromatic number of the complementary graph. This fact is called the sandwich theorem. In this paper, using new descriptions of the Lovász number and linear algebraic lemmas we give three proofs for a weaker version of the sandwich theorem.