A note on the /spl theta/ number of Lovasz and the generalized Delsarte bound

The /spl theta/ number of Lovasz and the /spl theta//sub 1/2/ of Schrijver, McEliece, Rodemich and Rumsey are convex (semidefinite) programming upper bounds on /spl alpha/(G), the size of a maximal independent set of G. It is known that /spl alpha/(G)/spl les//spl theta//sub 1/2/(G)/spl les//spl theta/(G)/spl les//spl chi/~(G), where /spl chi/~(G) is the clique cover number of G. The above inequalities suggest that perhaps /spl theta//sub 1/2/(G) approximates /spl alpha/(G) from above, and /spl theta/(G) approximates /spl chi/~(G) from below for every graph G. Can this approximation be to within a factor of at most n/sup 1-/spl epsiv// for some fixed /spl epsiv/>0? We show, that the following three conjectures are equivalent: 1. /spl exist//spl epsiv/>0 : /spl theta/(G) approximates /spl alpha/(G) for every G within a factor of n/sup n-/spl epsiv// 2. /spl exist//spl epsiv/>0 : /spl theta/(G) approximates /spl chi/~(G) for every G within a factor of n/sup n-/spl epsiv// 3. /spl exist//spl epsiv/>0 : /spl theta//sub 1/2/(G) approximates /spl alpha/(G) for every G within a factor of n/sup n-/spl epsiv// It is not impossible that /spl theta//sub 1/2/ approximates /spl chi/~(G), but the latter conjecture looks strictly stronger than 1-3. We give however a simple combinatorial reformulation of this one (we cannot find such for 1-3). We rule out some likely candidates for counterexamples to 1-3 by showing that /spl theta/(G) approximates /spl alpha/(G) and /spl chi/~(G) for those graphs G that come from the Hamming scheme.<<ETX>>