Complete characterization of graphs with local total antimagic chromatic number 3
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910507168956416 |
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| author | Lau, G. C. |
| author_facet | Lau, G. C. |
| contents | 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 incident vertex and edge have distinct induced weights where the induced weight of a vertex $v$, $w_f(v) = \sum f(e)$ 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 $χ_{lt}(G)$, is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of $G$. In this paper, we first obtained general lower and upper bounds for $χ_{lt}(G)$ and sufficient conditions to construct a graph $H$ with $k$ pendant edges and $χ_{lt}(H) \in\{Δ(H)+1, k+1\}$. We then completely characterized graphs $G$ with $χ_{lt}(G)=3$. Many families of (disconnected) graphs $H$ with $k$ pendant edges and $χ_{lt}(H) \in\{Δ(H)+1, k+1\}$ are also obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14653 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complete characterization of graphs with local total antimagic chromatic number 3 Lau, G. C. Combinatorics 05C78, 05C69 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 incident vertex and edge have distinct induced weights where the induced weight of a vertex $v$, $w_f(v) = \sum f(e)$ 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 $χ_{lt}(G)$, is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of $G$. In this paper, we first obtained general lower and upper bounds for $χ_{lt}(G)$ and sufficient conditions to construct a graph $H$ with $k$ pendant edges and $χ_{lt}(H) \in\{Δ(H)+1, k+1\}$. We then completely characterized graphs $G$ with $χ_{lt}(G)=3$. Many families of (disconnected) graphs $H$ with $k$ pendant edges and $χ_{lt}(H) \in\{Δ(H)+1, k+1\}$ are also obtained. |
| title | Complete characterization of graphs with local total antimagic chromatic number 3 |
| topic | Combinatorics 05C78, 05C69 |
| url | https://arxiv.org/abs/2401.14653 |