Radio Labeling of Supersub—Division of Path Graphs

Motivated by the channel assignment problem, we study the radio labeling of graphs. The radio labeling problem is an important topic in discrete mathematics due to its diverse applications, e.g., frequency assignment in mobile communication systems, signal processing, circuit and sensor network design, etc. A graph labeling problem is an assignment of labels to the vertices or edges (or both) of a graph G that satisfy a mathematical constraint. Radio labeling, a vertex labeling of graphs with non-negative integers, finds an important application in the study of radio channel assignment problems. The maximum label used in a radio labeling is called its span, and the smallest possible span of a radio labeling is called the radio number of a graph. In this area, Chartrand et al. (2005) provided important results by computing the exact values of
$rn(G)$
for paths and cycles when
$k$
is equal to the diameter for certain cases. In this paper, we determine the radio number
$rn(G)$
of
$G$
where
$G$
is the supersub−division of a path
$P_{n}$
with
$n \geq 3$
vertices and a complete bipartite graph
$K_{2, \alpha }$
with
$\alpha \in \mathbb {N}$
.