Proof of a local antimagic conjecture
An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two s...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2018-06-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3887/pdf |
| _version_ | 1827323763135348736 |
|---|---|
| author | John Haslegrave |
| author_facet | John Haslegrave |
| author_sort | John Haslegrave |
| collection | DOAJ |
| description | An antimagic labelling of a graph $G$ is a bijection
$f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$
distinguish all vertices. A well-known conjecture of Hartsfield and Ringel
(1994) is that every connected graph other than $K_2$ admits an antimagic
labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \&
Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \&
Lyngsie (2017)) independently introduced the weaker notion of a local antimagic
labelling, where only adjacent vertices must be distinguished. Both sets of
authors conjectured that any connected graph other than $K_2$ admits a local
antimagic labelling. We prove this latter conjecture using the probabilistic
method. Thus the parameter of local antimagic chromatic number, introduced by
Arumugam et al., is well-defined for every connected graph other than $K_2$ . |
| first_indexed | 2024-04-25T01:58:48Z |
| format | Article |
| id | doaj.art-b156c70281ef49cdb3cb01797af39cb8 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T01:58:48Z |
| publishDate | 2018-06-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-b156c70281ef49cdb3cb01797af39cb82024-03-07T15:36:37ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502018-06-01Vol. 20 no. 1Graph Theory10.23638/DMTCS-20-1-183887Proof of a local antimagic conjectureJohn HaslegraveAn antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .https://dmtcs.episciences.org/3887/pdfmathematics - combinatorics05c78, 05c15 |
| spellingShingle | John Haslegrave Proof of a local antimagic conjecture Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics 05c78, 05c15 |
| title | Proof of a local antimagic conjecture |
| title_full | Proof of a local antimagic conjecture |
| title_fullStr | Proof of a local antimagic conjecture |
| title_full_unstemmed | Proof of a local antimagic conjecture |
| title_short | Proof of a local antimagic conjecture |
| title_sort | proof of a local antimagic conjecture |
| topic | mathematics - combinatorics 05c78, 05c15 |
| url | https://dmtcs.episciences.org/3887/pdf |
| work_keys_str_mv | AT johnhaslegrave proofofalocalantimagicconjecture |