Affirmative Solutions on Local Antimagic Chromatic Number
An edge labeling of a connected graph G = ( V , E ) is said to be local antimagic if it is a bijection f : E → { 1 , … , | E | } such that for any pair of adjacent vertices x and y , f + ( x ) ≠ f + ( y ) , where the induced vertex label f + ( x ) = ∑ f ( e ) , with e ranging over all the edges inci...
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| Published in: | Graphs and combinatorics Vol. 36; no. 5; pp. 1337 - 1354 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Tokyo
Springer Japan
01.09.2020
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0911-0119, 1435-5914 |
| Online Access: | Get full text |
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| Summary: | An edge labeling of a connected graph
G
=
(
V
,
E
)
is said to be local antimagic if it is a bijection
f
:
E
→
{
1
,
…
,
|
E
|
}
such that for any pair of adjacent vertices
x
and
y
,
f
+
(
x
)
≠
f
+
(
y
)
, where the induced vertex label
f
+
(
x
)
=
∑
f
(
e
)
, with
e
ranging over all the edges incident to
x
. The local antimagic chromatic number of
G
, denoted by
χ
la
(
G
)
, is the minimum number of distinct induced vertex labels over all local antimagic labelings of
G
. In this paper, we give counterexamples to the lower bound of
χ
la
(
G
∨
O
2
)
that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of
χ
la
(
G
∨
O
n
)
and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-020-02197-2 |