Improved upper bounds for nearly antipodal chromatic number of paths

For paths Pn, G. Chartrand, L. Nebeský and P. Zhang showed that ac′(Pn) ≤ ( n−2 2 ) +2 for every positive integer n, where ac′(Pn) denotes the nearly antipodal chromatic number of Pn. In this paper we show that ac′(Pn) ≤ ( n−2 2 ) − n2 − b 10 n c + 7 if n is even positive integer and n ≥ 10, and ac′(Pn) ≤ ( n−2 2 ) − n−1 2 − b 13 n c + 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.