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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2403.01492 |
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| _version_ | 1866909126098944000 |
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| author | Wu, T. Luo, J. Gao, Y. |
| author_facet | Wu, T. Luo, J. Gao, Y. |
| contents | A graph $G=(V,E)$ is called $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a mapping $f:V\cup E\rightarrow \mathbb{R}$ such that $f(y)\in L(y)$ for any $y\in V\cup E$ and for any two adjacent vertices $v, v'$, $\sum_{e\in E(v)}f(e)+f(v)\neq \sum_{e\in E(v')}f(e)+f(v')$, where $E(x)$ denotes the set of incident edges of a vertex $x\in V(G)$. In this paper, we characterize a sufficient condition on
$(1,2)$-choosable of graphs. We show that every connected $(n,m)$-graph is both $(2,2)$-choosable and $(1,3)$-choosable if $m=n$ or $n+1$, where $(n,m)$-graph denotes the graph with $n$ vertices and $m$ edges. Furthermore, we prove that some graphs obtained by some graph operations are $(2,2)$-choosable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some results on total weight choosability Wu, T. Luo, J. Gao, Y. Combinatorics A graph $G=(V,E)$ is called $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a mapping $f:V\cup E\rightarrow \mathbb{R}$ such that $f(y)\in L(y)$ for any $y\in V\cup E$ and for any two adjacent vertices $v, v'$, $\sum_{e\in E(v)}f(e)+f(v)\neq \sum_{e\in E(v')}f(e)+f(v')$, where $E(x)$ denotes the set of incident edges of a vertex $x\in V(G)$. In this paper, we characterize a sufficient condition on $(1,2)$-choosable of graphs. We show that every connected $(n,m)$-graph is both $(2,2)$-choosable and $(1,3)$-choosable if $m=n$ or $n+1$, where $(n,m)$-graph denotes the graph with $n$ vertices and $m$ edges. Furthermore, we prove that some graphs obtained by some graph operations are $(2,2)$-choosable. |
| title | Some results on total weight choosability |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.01492 |