Shannon capacity, Lov\'asz theta number and the Mycielski construction

TL;DR

This paper explores how the Mycielski construction affects the Shannon capacity and Lovász theta number of graphs, revealing that it can increase capacity and that the theta number's value is determined by the original graph.

Contribution

It demonstrates that the Mycielski construction can increase Shannon capacity and characterizes the Lovász theta number's behavior on Mycielski graphs, extending understanding of graph invariants.

Findings

Mycielski construction can increase Shannon capacity when capacity is attained by finite powers.

Lovász theta number on Mycielski graphs is determined by the original graph's theta number.

Results suggest potential generalizations and connections to the asymptotic spectrum of graphs.

Abstract

We investigate the effect of the well-known Mycielski construction on the Shannon capacity of graphs and on one of its most prominent upper bounds, the (complementary) Lov\'asz theta number. We prove that if the Shannon capacity of a graph, the distinguishability graph of a noisy channel, is attained by some finite power, then its Mycielskian has strictly larger Shannon capacity than the graph itself. For the complementary Lov\'asz theta function we show that its value on the Mycielskian of a graph is completely determined by its value on the original graph, a phenomenon similar to the one discovered for the fractional chromatic number by Larsen, Propp and Ullman. We also consider the possibility of generalizing our results on the Sperner capacity of directed graphs and on the generalized Mycielsky construction. Possible connections with what Zuiddam calls the asymptotic spectrum of graphs are discussed as well.