Complete characterization of graphs with local total antimagic chromatic number 3
A total labeling of a graph \(G = (V, E)\) is said to be local total antimagic if it is a bijection \(f: V\cup E \to\{1,\ldots,|V|+|E|\}\) such that adjacent vertices, adjacent edges, and pairs of an incident vertex and edge have distinct induced weights where the induced weight of a vertex \(v\) is...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AGH Univeristy of Science and Technology Press
2025-03-01
|
| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4511.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850033180431613952 |
|---|---|
| author | Gee-Choon Lau |
| author_facet | Gee-Choon Lau |
| author_sort | Gee-Choon Lau |
| collection | DOAJ |
| description | A total labeling of a graph \(G = (V, E)\) is said to be local total antimagic if it is a bijection \(f: V\cup E \to\{1,\ldots,|V|+|E|\}\) such that adjacent vertices, adjacent edges, and pairs of an incident vertex and edge have distinct induced weights where the induced weight of a vertex \(v\) is \(w_f(v) = \sum f(e)\) with \(e\) ranging over all the edges incident to \(v\), and the induced weight of an edge \(uv\) is \(w_f(uv) = f(u) + f(v)\). The local total antimagic chromatic number of \(G\), denoted by \(\chi_{lt}(G)\), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of \(G\). In this paper, we first obtain general lower and upper bounds for \(\chi_{lt}(G)\) and sufficient conditions to construct a graph \(H\) with \(k\) pendant edges and \(\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}\). We then completely characterize graphs \(G\) with \(\chi_{lt}(G)=3\). Many families of (disconnected) graphs \(H\) with \(k\) pendant edges and \(\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}\) are also obtained. |
| format | Article |
| id | doaj-art-8cf2ac256bd74a35956dde969c291a7d |
| institution | DOAJ |
| issn | 1232-9274 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj-art-8cf2ac256bd74a35956dde969c291a7d2025-08-20T02:58:18ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742025-03-01452199225https://doi.org/10.7494/OpMath.2025.45.2.1994511Complete characterization of graphs with local total antimagic chromatic number 3Gee-Choon Lau0https://orcid.org/0000-0002-9777-657177D, Jalan Subuh, 85000, Johor, MalaysiaA total labeling of a graph \(G = (V, E)\) is said to be local total antimagic if it is a bijection \(f: V\cup E \to\{1,\ldots,|V|+|E|\}\) such that adjacent vertices, adjacent edges, and pairs of an incident vertex and edge have distinct induced weights where the induced weight of a vertex \(v\) is \(w_f(v) = \sum f(e)\) with \(e\) ranging over all the edges incident to \(v\), and the induced weight of an edge \(uv\) is \(w_f(uv) = f(u) + f(v)\). The local total antimagic chromatic number of \(G\), denoted by \(\chi_{lt}(G)\), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of \(G\). In this paper, we first obtain general lower and upper bounds for \(\chi_{lt}(G)\) and sufficient conditions to construct a graph \(H\) with \(k\) pendant edges and \(\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}\). We then completely characterize graphs \(G\) with \(\chi_{lt}(G)=3\). Many families of (disconnected) graphs \(H\) with \(k\) pendant edges and \(\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}\) are also obtained.https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4511.pdflocal total antimagiclocal total antimagic chromatic number |
| spellingShingle | Gee-Choon Lau Complete characterization of graphs with local total antimagic chromatic number 3 Opuscula Mathematica local total antimagic local total antimagic chromatic number |
| title | Complete characterization of graphs with local total antimagic chromatic number 3 |
| title_full | Complete characterization of graphs with local total antimagic chromatic number 3 |
| title_fullStr | Complete characterization of graphs with local total antimagic chromatic number 3 |
| title_full_unstemmed | Complete characterization of graphs with local total antimagic chromatic number 3 |
| title_short | Complete characterization of graphs with local total antimagic chromatic number 3 |
| title_sort | complete characterization of graphs with local total antimagic chromatic number 3 |
| topic | local total antimagic local total antimagic chromatic number |
| url | https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4511.pdf |
| work_keys_str_mv | AT geechoonlau completecharacterizationofgraphswithlocaltotalantimagicchromaticnumber3 |