Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach
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| Format: | Preprint |
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2024
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| author | Lau, Gee-Choon Shiu, Wai Chee Nalliah, M. Premalatha, K. |
| author_facet | Lau, Gee-Choon Shiu, Wai Chee Nalliah, M. Premalatha, K. |
| contents | An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it 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 $G$. Suppose $χ_{la}(G)=χ_{la}(H)$ and $G_H$ is obtained from $G$ and $H$ by merging some vertices of $G$ with some vertices of $H$ bijectively. In this paper, we give ways to construct matrices with integers in $[1,10k]$, $k\ge 1$, that meet certain properties. Consequently, we obtained many families of (disconnected) bipartite (and tripartite) graphs of size $10k$ with local antimagic chromatic number 3. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_18049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach Lau, Gee-Choon Shiu, Wai Chee Nalliah, M. Premalatha, K. Combinatorics 05C78, 05C69 An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it 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 $G$. Suppose $χ_{la}(G)=χ_{la}(H)$ and $G_H$ is obtained from $G$ and $H$ by merging some vertices of $G$ with some vertices of $H$ bijectively. In this paper, we give ways to construct matrices with integers in $[1,10k]$, $k\ge 1$, that meet certain properties. Consequently, we obtained many families of (disconnected) bipartite (and tripartite) graphs of size $10k$ with local antimagic chromatic number 3. |
| title | Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach |
| topic | Combinatorics 05C78, 05C69 |
| url | https://arxiv.org/abs/2404.18049 |