NEARLY ANTIPODAL CHROMATIC NUMBER ac′(Pn) OF THE PATH Pn

Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number ac′(Pn) as (n−2 2 ) +2 for n > 9 and have found the exact value of ac′(Pn) for n = 5, 6, 7, 8. Here we determine the exact values of ac′(Pn) for n > 8. They are 2p − 6p + 8 for n = 2p and 2p − 4p + 6 for n = 2p + 1. The exact value of the radio antipodal number ac(Pn) for the path Pn of order n has been determined by Khennoufa and Togni in 2005 as 2p− 2p+3 for n = 2p+1 and 2p− 4p+5 for n = 2p. Although the value of ac(Pn) determined there is correct, we found a mistake in the proof of the lower bound when n = 2p (Theorem 6). However, we give an easy observation which proves this lower bound.