Abstract
Suppose that
G
is a connected graph. For any two vertices
u
and
v
, let
d
G
(u,v) denote the distance between
u
and
v
in
G
. The diameter of
G
is the maximum distance between any pair of vertices, and it is denoted by diam(
G
). A multilevel distance labeling (or radio condition) for
G
is a function
f
that assigns to each vertex of
G
a positive integer such that for any distinct vertices
u
and
v
,
d
(
u
,
v
) + |
f
(
u
) −
f
(
v
)| ≥ diam(
G
) + 1. The largest positive integer in the range of
f
is called the span of
f
. The radio number of
G
is denoted by
r
n
(
G
), is the minimum span of a multilevel distance labeling for
G
. In this paper the multilevel distance labeling of
H
n
is calculated for every
n
≥ 15:
r
n
(
H
n
) = 4
n
+ 2.