On Radio (n — 4)-chromatic Number of the Path P_n

Abstract

For a path P_n 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 P_n, namely radio number, rn(P_n), when n ≥ 3. Khennoufa and Togni [5] have given the exact value of radio (n — 2)-chromatic number of P_n, namely antipodal number, ac(P_n), when n ≥ 5. Kola and Panigrahi [6] have given the exact value of radio (n — 3)-chromatic number of P_n, namely nearly antipodal number, ac'(P_n), when n ≥ 8. In this paper, we give the exact value of radio (n — 4)-chromatic number of P_n, rc_n-4(P_n), when n is odd and n ≥ 11. Consequently, the lower bound of rc_n-4(P_n+i), n ≥ 11 and i ≥ 1 is improved. We also improve the upper bound of rc_n-4(P_n) when n is even and n ≥ 12.

Keywords:

• radio k-coloring
• Span of a radio k-coloring
• Radio k-chromatic number

2000 Mathematics Subject Classification:

• 05C15
• 05C78
• 05C12