Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach
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arXiv
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| Format: | Preprint |
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2024
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| author | Lau, Gee-Choon Shiu, Wai Chee Premalatha, K. Nalliah, M. |
| author_facet | Lau, Gee-Choon Shiu, Wai Chee Premalatha, K. Nalliah, M. |
| contents | An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if there is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, 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 three ways to construct a $(3m+2)\times (2k+1)$ matrix that meets certain properties for $m=1,3$ and $k\ge 1$. Consequently, we obtained many (disconnected) graphs of size $(3m+2)(2k+1)$ with local antimagic chromatic number 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16484 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach Lau, Gee-Choon Shiu, Wai Chee Premalatha, K. Nalliah, M. Combinatorics 05C78, 05C69 An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if there is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, 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 three ways to construct a $(3m+2)\times (2k+1)$ matrix that meets certain properties for $m=1,3$ and $k\ge 1$. Consequently, we obtained many (disconnected) graphs of size $(3m+2)(2k+1)$ with local antimagic chromatic number 3. |
| title | Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach |
| topic | Combinatorics 05C78, 05C69 |
| url | https://arxiv.org/abs/2403.16484 |