The Fractional Haemers Bound of The Mycielski Construction

We investigate the effect of the generalized Mycielski construction Mᵣ(G) on the complementary fractional Haemers bound H̄f(G; F), a parameter that depends on a graph G and a field F. The effect of the Mycielski construction on graph parameters has already been studied for the fractional chromatic number χf and the complementary Lovász theta number ϑ̄. Larsen, Propp, and Ullman provided a formula for χf(M₂(G)) in terms of χf(G). This was later generalized by Tardif to χf(Mᵣ(G)) for any r, and Simonyi and the author gave a similar expression for ϑ̄(M₂(G)) in terms of ϑ̄(G). In this paper, we show that Tardif's formula for the fractional chromatic number remains valid for H̄f whenever H̄f(G; F) equals the clique number of G. In particular, we provide a general upper bound on H̄f(Mᵣ(G); F) in terms of H̄f(G;F) and we prove that this bound is tight whenever H̄f(G; F) equals the clique number of G.