On the Radio  -chromatic Number of Paths

doi:10.1285/i15900932v42n1p37

On the Radio $k$ -chromatic Number of Paths

Niranjan P. K., Srinivasa Rao Kola

Abstract

A radio $k$ -coloring of a graph $G$ is an assignment $f$ of positive integers (colors) to the vertices of $G$ such that for any two vertices $u$ and $v$ of $G$ , the difference between their colors is at least $1+k-d(u,v)$ . The span $rc_k(f)$ of $f$ is $\max\{f(v):v\in V(G)\}$ . The radio $k$ -chromatic number $rc_k(G)$ of $G$ is $min\lbrace rc_k(f) : f { is a radio k\text{-}coloring of } G\rbrace$ . In this paper, in an attempt to prove a conjecture on the radio $k$ -chromatic number of path, we determine the radio $k$ -chromatic number of paths $P_n$ for $k+5\leq n\leq\frac{7k-1}{2}$ if $k$ is odd and $k+4\leq n\leq\frac{5k+4}{2}$ if $k$ is even.

DOI Code: 10.1285/i15900932v42n1p37

Keywords: radio k-coloring; radio k-chromatic number; radio coloring; radio number

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On the Radio -chromatic Number of Paths

Abstract

On the Radio $k$ -chromatic Number of Paths