Abstract
AbstractThe circular $$L(j,k)$$
L
(
j
,
k
)
-labeling problem with $$k\ge j$$
k
≥
j
arose from the code assignment in the wireless network of computers. Given a graph $$G$$
G
and positive numbers $$j,k,\sigma $$
j
,
k
,
σ
, and a circular $$\sigma $$
σ
-$$L(j,k)$$
L
(
j
,
k
)
-labeling of a graph $$G$$
G
is an assignment $$f$$
f
from $$[0,\sigma )$$
[
0
,
σ
)
to the vertices of $$G$$
G
, for any two vertices $$u$$
u
and $$v$$
v
, such that $$|f(u)-f(v){|}_{\sigma }\ge j$$
|
f
(
u
)
-
f
(
v
)
|
σ
≥
j
if $$uv\in E(G)$$
u
v
∈
E
(
G
)
, and $$|f(u)-f(v){|}_{\sigma }\ge k$$
|
f
(
u
)
-
f
(
v
)
|
σ
≥
k
if $$u$$
u
and $$v$$
v
are distance two apart, where $$|f\left(u\right)-f\left(v\right){|}_{\sigma }=min\left\{|f(u)-f(v)|, \sigma -|f(u)-f(v)|\right\}$$
|
f
u
-
f
v
|
σ
=
m
i
n
|
f
(
u
)
-
f
(
v
)
|
,
σ
-
|
f
(
u
)
-
f
(
v
)
|
. The minimum $$\sigma $$
σ
such that graph $$G$$
G
has a circular $$\sigma $$
σ
-$$L(j,k)$$
L
(
j
,
k
)
-labeling of a graph $$G$$
G
, which is called the circular $$L(j,k)$$
L
(
j
,
k
)
-labeling number of graph $$G$$
G
and is denoted by $${\sigma }_{j,k}(G)$$
σ
j
,
k
(
G
)
. In this paper, we determine the circular $$L(j,k)$$
L
(
j
,
k
)
-labeling numbers of Cartesian product of three paths, where $$k\ge 2j.$$
k
≥
2
j
.
Publisher
Springer Nature Singapore