An improved lower bound for the packing number of the 2-token graph of the cycle

Abstract

Let $F_2(C_n)$ be the $2$-token graph of the cycle $C_n$ and let $\rho$ denote the packing number. Gómez Soto and Ríos-Castro recently proved that $\rho (F_2(C_n))\ge a(n)$ for $n\ge 19$, where $a(n)$ is an explicit expression. In this note, we prove that \[ ρ(F_2(C_n))\ \ge\ \left\lfloor\frac{n(n-2)}{10}\right\rfloor+1\qquad\text{for every } n\ge 3, \] which improves $a(n)$ by one whenever $n\equiv 0,2(\mathrm{mod}10)$.

arXiv comment9 pages, preliminary version, The AI assistant Claude (Antropic) was used at some stages of the project