On Radio k-Chromatic Number of Various Graphs

Objective: The main objective of the study is to reduce radio spectrums without any interference. This study aims to find the smallest span of radio k chromatic number of various graphs. The minimal number of colors essential to color a graph is called its span. Methods: In this paper, we address the issue of minimizing interference by modeling it as a radio k-coloring problem on graphs, when vertices and the connections between them represent the spectrum bandwidth indicated by edges. For a positive integer k, a radio k – coloring of simply connected graph T, is a function 𝜙: V(T) →{c0, c1, c2, . . .} such that |𝜙 (𝑢) − 𝜙 (𝑣)| ≥ 1 + 𝑘 − 𝑑(𝑢, 𝑣) for each pair of distinct vertices u and v of T, when diam(𝑢, 𝑣) is the distance between u and v in T. The maximum color granted by 𝜙 is called span, which can be denoted by 𝑟𝑐𝑘(𝜙). The radio k – chromatic number 𝑟𝑐𝑘(T) of T is the minimal {rc𝑘(𝜙)}, when 𝜙 is a radio k -coloring of T. Findings: This study presents the exact value of the radio kchromatic number of some graphs like T𝑛, TC𝑛, A(T𝑛), U𝑛,𝑛−1 for 2 ≤ 𝑘 ≤ diam(T), when diam(T) is the diameter of graph T. Novelty: By introducing an interference graph, the channel assigned gets transformed into a graph coloring issue, eventually confronting the channel assigned problem for radio spectrum. Keywords: Radio k – Chromatic Number; Triangular Snake Graph; Triangular Cactus Graph; Alternative Triangular Snake Graph; Umbrella Graph