An example showing that Schrijver's $\vartheta$-function need not upper bound the Shannon capacity of a graph

TL;DR

This paper provides a concrete example demonstrating that Schrijver's $ heta$-function does not always serve as an upper bound for the Shannon capacity of a graph, clarifying a long-standing open question.

Contribution

It presents the first explicit Tanner graph example showing Schrijver's $ heta$-function can fail to upper bound Shannon capacity, resolving an open problem.

Findings

Schrijver's $ heta$-function does not always upper bound Shannon capacity

Explicit Tanner graph example with 32 vertices

Clarifies the distinction between Schrijver's and Lovász's $ heta$ functions

Abstract

This letter addresses an open question concerning a variant of the Lov\'{a}sz $\vartheta$ function, which was introduced by Schrijver and independently by McEliece et al. (1978). The question of whether this variant provides an upper bound on the Shannon capacity of a graph was explicitly stated by Bi and Tang (2019). This letter presents an explicit example of a Tanner graph on 32 vertices, which shows that, in contrast to the Lov\'{a}sz $\vartheta$ function, this variant does not necessarily upper bound the Shannon capacity of a graph. The example, previously outlined by the author in a recent paper (2024), is presented here in full detail, making it easy to follow and verify. By resolving this question, the note clarifies a subtle but significant distinction between these two closely related graph invariants.