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Recurrence for v-number of binary trees

Establish the recurrence v(J_{B_n}) = 2^{n−1} + v(J_{B_{n−3}}) for all n ≥ 3, where J_{B_n} is the binomial edge ideal of the binary tree B_n of level n (with 2^{n+1}−1 vertices).

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Background

The authors derive sharp upper bounds for v(J_{B_n}) and suggest, based on structural analysis and computations, that the v-number follows a specific recurrence in n. This recurrence relates the growth in v-number to the combinatorial expansion of the tree.

Confirming this recurrence would provide a clean inductive description of v-numbers for binary trees and complement exact formulas known for paths and conjectured 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)