Improved upper bounds for nearly antipodal chromatic number of paths

Abstract

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