ON RADIO $k$-CHROMATIC NUMBERS FOR THE FAMILY OF TRIPLE STAR GRAPHS

Abstract

Let $G=\{V(G), E(G)\}$ be a simple connected graph with diameter $d(G)$ and $k$ be a positive integer. A radio $k$-coloring of a graph is a proper node coloring that is an assignment $f$ of positive integers to the nodes of $G$ such that $|f(x)-f(y)| \geq k+1-d(x, y)$, where $x$ and $y$ are two distinct nodes, and $d(x, y)$ is the length between $x$ and $y$. The maximum color assigned to some node of $f(V(G))$ is called the span of $f$ and it is indicated by $\operatorname{span}(f)$. The least span over all radio $k$-coloring of $G$ is the radio $k$-chromatic number of $G$ and it is indicated by $r c_{\hbar}(G)$. In this paper, we investigate the radio $k$-chromatic number for the triple star graph $K_{1, n, n, n}$ and its middle graph $M\left(K_{1, n, n, n}\right)$, central graph $C\left(K_{1, n, n, n}\right)$, total graph $T\left(K_{1, n, n, n}\right)$ and line graph $L\left(K_{1, n, n, n}\right)$.

Received: October 4, 2023
Revised: November 22, 2023
Accepted: February 6, 2024