On Radio ( n − 4 ) - chromatic Number of the Path P n

For a path Pn of order n, Chartrand et al. [3] have given an upper bound for radio k-chromatic number when 1 ≤ k ≤ n−1. Liu and Zhu [7] have determined the exact value of radio (n−1)-chromatic number of Pn, namely radio number, rn(Pn), when n ≥ 3. Khennoufa and Togni [5] have given the exact value of radio (n − 2)-chromatic number of Pn, namely antipodal number, ac(Pn), when n ≥ 5. Kola and Panigrahi [6] have given the exact value of radio (n−3)-chromatic number of Pn, namely nearly antipodal number, ac′(Pn), when n ≥ 8. In this paper, we give the exact value of radio (n − 4)-chromatic number of Pn, rcn−4(Pn), when n is odd and n ≥ 11. Consequently, the lower bound of rcn−4(Pn+i), n ≥ 11 and i ≥ 1 is improved. We also improve the upper bound of rcn−4(Pn) when n is even and n ≥ 12.