Geodesic
Improved upper bounds for nearly antipodal chromatic number of paths
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 159-174.

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For paths Pₙ, G. Chartrand, L. Nebeský and P. Zhang showed that ac'(Pₙ) ≤ n-22 + 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-22 - n/2 - ⎣10/n⎦ + 7 if n is even positive integer and n ≥ 10, and ac'(Pₙ) ≤ n-22 - (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 Pₙ.
Keywords: radio colorings, nearly antipodal chromatic number, paths 
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Shen, Yu-Fa; Zheng, Guo-Ping; He, Wen-Jie. Improved upper bounds for nearly antipodal chromatic number of paths. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 159-174. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a13/

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