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arXiv · 2505.07778

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

Igal Sason
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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.

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Igal Sason. 2025-05-12. An example showing that Schrijver's $\vartheta$-function need not upper bound the Shannon capacity of a graph. https://doi.org/10.3934/math.2025685

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