Quantum independence and chromatic numbers

Abstract

In this thesis we are studying the cases when quantum independence and quantum chromatic numbers coincide with or differ from their classical counterparts. Knowing about the relation of chromatic numbers separation to the projective Kochen-Specker sets, we found an analogous characterisation for the independence numbers case. Additionally, all the graphs that we studied that had known quantum parameters exhibited both the separation between the classical and quantum independence numbers and the separation between the classical and quantum chromatic numbers. This observation and the Kochen-Specker connection suggested the possibility of the chromatic and independence numbers separations occurring simultaneously. We have disproved this idea with a counterexample. Furthermore, we generalised Manĉinska-Roberson’s example of the chromatic numbers separation to an infinite family. We investigate some known instances with strictly larger quantum independence numbers in-depth, find a more general description and generalise Piovesan’s example. Using the Lovász theta bound, we prove that there is no separation between the independence numbers in bipartite and perfect graphs. We also show that there is no separation when the classical independence number is two; and that the cone over a graph has the same quantum independence number as the underlying graph.