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Exact v-number for cycles

Determine the exact value of v(J_{C_n}) for the binomial edge ideal J_{C_n} of the cycle graph C_n on n vertices, by proving that v(J_{C_n}) = n - ⌊n/3⌋ = ⌈2n/3⌉ for all n ≥ 6.

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Background

The authors provide sharp upper bounds for the v-number of cycle graphs and present computational evidence suggesting a precise formula for large cycles. They conjecture a specific closed-form expression consistent with their bounds and examples.

Resolving this would extend the understanding of v-numbers beyond paths and Cohen–Macaulay closed graphs and provide a tight combinatorial characterization for cycles.

References

Conjecture {\em Let $C_n$ denote the cycle graph on $n$ vertices and $B_n$ denote the binary tree of level $n$. Then (1) $\mathrm{v}(J_{C_n}) = n - \lfloor\frac{n}{3} \rfloor=\lceil \frac{2n}{3}\rceil$ for all $n \geq 6$; (2) $\mathrm{v}(J_{B_n}) = 2{n-1} + \mathrm{v}(J_{B_{n-3})$ for all $n \geq 3$.}

— On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs (2405.15354 - Dey et al., 24 May 2024) in Section 3 (v=2 and expected v-number of cycles and binary trees)