Let
be a connected graph.
For any two vertices
and
, let
denote the
distance between
and
in
.
The maximum distance between any pair of vertices of
is called the
diameter of
and
denoted by
.
A
radio labeling (or multilevel distance labeling) of
is a
function
that assigns to each vertex a label from the set
such that the following
holds for any vertices
and
:
. The
span of
is defined as
. The
radionumber of
is the minimum span over all radio labelings of
. The
fourth power
of
is a graph
constructed from
by adding edges between vertices of distance four or less apart in
.
In this paper, we completely determine the radio number for the
fourth power of any path, except when its order is congruent to
.
Keywords
channel assignment problem, multilevel distance labeling,
radio number, radio labeling
