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A New Approach in Finding Full Friendly Indices
Let G be a graph with vertex set V ( G ) and edge set E ( G ), a vertex labeling f : V ( G ) → Z 2 induces an edge labeling f + : E ( G ) → Z 2 defined by f + ( x y ) = f ( x ) + f ( y ) , for each edge x y ∈ E ( G ) . For each i ∈ Z 2 , let v f ( i ) = | { u ∈ V ( G ) : f ( u ) = i } | and e f + (...
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| Published in: | Bulletin of the Malaysian Mathematical Sciences Society 2018, Vol.41 (1), p.443-453 |
|---|---|
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Let
G
be a graph with vertex set
V
(
G
) and edge set
E
(
G
), a vertex labeling
f
:
V
(
G
)
→
Z
2
induces an edge labeling
f
+
:
E
(
G
)
→
Z
2
defined by
f
+
(
x
y
)
=
f
(
x
)
+
f
(
y
)
, for each edge
x
y
∈
E
(
G
)
. For each
i
∈
Z
2
, let
v
f
(
i
)
=
|
{
u
∈
V
(
G
)
:
f
(
u
)
=
i
}
|
and
e
f
+
(
i
)
=
|
{
x
y
∈
E
(
G
)
:
f
+
(
x
y
)
=
i
}
|
. A vertex labeling
f
of a graph
G
is said to be friendly if
|
v
f
(
1
)
-
v
f
(
0
)
|
≤
1
. The friendly index set of the graph
G
, denoted by
FI
(
G
), is defined as
{
|
e
f
+
(
1
)
-
e
f
+
(
0
)
|
: the vertex labeling
f
is friendly
}
. The full friendly index set of the graph
G
, denoted by
FFI
(
G
), is defined as
{
e
f
+
(
1
)
-
e
f
+
(
0
)
: the vertex labeling
f
is friendly
}
. In this paper, we determine
FFI
(
G
) for a class of cubic graphs with full vertices blow-up of cycle by a complete tripartite graph
K
(1, 1, 2) using a new method known as embedding labeling graph method. As a by-product, we also discuss the cordiality and the full product-cordial index sets for this graph. |
|---|---|
| ISSN: | 0126-6705 2180-4206 |
| DOI: | 10.1007/s40840-016-0373-8 |