On Bounds for the Product Irregularity Strength of Graphs
For a graph X with at most one isolated vertex and without isolated edges, a product-irregular labeling ω : E ( X ) → { 1 , 2 , … , s } , first defined by Anholcer in 2009, is a labeling of the edges of X such that for any two distinct vertices u and v of X the product of labels of the edges inciden...
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| Published in | Graphs and combinatorics Vol. 31; no. 5; pp. 1347 - 1357 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
Springer Japan
01.09.2015
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | For a graph
X
with at most one isolated vertex and without isolated edges, a
product-irregular labeling
ω
:
E
(
X
)
→
{
1
,
2
,
…
,
s
}
, first defined by Anholcer in 2009, is a labeling of the edges of
X
such that for any two distinct vertices
u
and
v
of
X
the product of labels of the edges incident with
u
is different from the product of labels of the edges incident with
v
. The minimal
s
for which there exist a product irregular labeling is called
the product irregularity strength
of
X
and is denoted by
p
s
(
X
)
. In this paper it is proved that
p
s
(
X
)
≤
|
V
(
X
)
|
-
1
for any graph
X
with more than
3
vertices. Moreover, the connection between the product irregularity strength and the multidimensional multiplication table problem is given, which is especially expressed in the case of the complete multipartite graphs. |
|---|---|
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-014-1458-5 |