Improved upper bounds for nearly antipodal chromatic number of paths

For paths Pₙ, G. Chartrand, L. Nebeský and P. Zhang showed that ${\displaystyle ac\prime (Pₙ)\le b\in om\{n-2\}\left\{2\right\}+2}$ for every positive integer n, where ac'(Pₙ) denotes the nearly antipodal chromatic number of Pₙ. In this paper we show that ${\displaystyle ac\prime (Pₙ)\le b\in om\{n-2\}\left\{2\right\}-\frac{n}{2}-⎣\frac{10}{n}⎦+7}$ if n is even positive integer and n ≥ 10, and ${\displaystyle ac\prime (Pₙ)\le b\in om\{n-2\}\left\{2\right\}-\frac{n-1}{2}-⎣\frac{13}{n}⎦+8}$ if n is odd positive integer and n ≥ 13. For all even positive integers n ≥ 10 and all odd positive integers n ≥ 13, these results improve the upper bounds for nearly antipodal chromatic number of Pₙ.