TY  - JOUR
AU  - Sason, Igal
PY  - 2025
TI  - An example showing that Schrijver's    ϑ-function need not upper bound the Shannon capacity of a graph
JO  - AIMS Mathematics
SP  - 15294
EP  - 15301
VL  - 10
IS  - 7
AB  - This letter addresses an open question concerning a variant of the Lovász    ϑ 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 Lovász    ϑ 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.
UR  - https://doi.org/10.3934/math.2025685
DO  - 10.3934/math.2025685