Approximate results for rainbow labelings
A simple graph G = ( V , E ) is said to be antimagic if there exists a bijection f : E → [ 1 , | E | ] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f : V → [ 1 , | V | ] ,...
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| Vydáno v: | Periodica mathematica Hungarica Ročník 74; číslo 1; s. 11 - 21 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article Publikace |
| Jazyk: | angličtina |
| Vydáno: |
Dordrecht
Springer Netherlands
01.03.2017
Springer Nature B.V |
| Témata: | |
| ISSN: | 0031-5303, 1588-2829 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | A simple graph
G
=
(
V
,
E
)
is said to be antimagic if there exists a bijection
f
:
E
→
[
1
,
|
E
|
]
such that the sum of the values of
f
on edges incident to a vertex takes different values on distinct vertices. The graph
G
is distance antimagic if there exists a bijection
f
:
V
→
[
1
,
|
V
|
]
,
such that
∀
x
,
y
∈
V
,
∑
x
i
∈
N
(
x
)
f
x
i
≠
∑
x
j
∈
N
(
y
)
f
x
j
.
Using the polynomial method of Alon we prove that there are antimagic injections of any graph
G
with
n
vertices and
m
edges in the interval
[
1
,
2
n
+
m
-
4
]
and, for trees with
k
inner vertices, in the interval
[
1
,
m
+
k
]
.
In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph
G
with order
n
and maximum degree
Δ
in the interval
[
1
,
n
+
t
(
n
-
t
)
]
,
where
t
=
min
{
Δ
,
⌊
n
/
2
⌋
}
,
and, for trees with
k
leaves, in the interval
[
1
,
3
n
-
4
k
]
.
In particular, all trees with
n
=
2
k
vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0031-5303 1588-2829 |
| DOI: | 10.1007/s10998-016-0151-2 |