Multigraph edge-coloring with local list sizes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915103387942912 |
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| author | Dhawan, Abhishek |
| author_facet | Dhawan, Abhishek |
| contents | Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a list assignment on the edges of $G$. Suppose additionally, for every vertex $x$, the edges incident to $x$ have at least $f(x)$ colors in common. We consider a variant of local edge-colorings wherein the color received by an edge $e$ must be contained in $L(e)$. The locality appears in the function $f$, i.e., $f(x)$ is some function of the local structure of $x$ in $G$. Such a notion is a natural generalization of traditional local edge-coloring. Our main results include sufficient conditions on the function $f$ to construct such colorings. As corollaries, we obtain local analogs of Vizing and Shannon's theorems, recovering a recent result of Conley, Grebík and Pikhurko. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12094 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multigraph edge-coloring with local list sizes Dhawan, Abhishek Combinatorics Discrete Mathematics Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a list assignment on the edges of $G$. Suppose additionally, for every vertex $x$, the edges incident to $x$ have at least $f(x)$ colors in common. We consider a variant of local edge-colorings wherein the color received by an edge $e$ must be contained in $L(e)$. The locality appears in the function $f$, i.e., $f(x)$ is some function of the local structure of $x$ in $G$. Such a notion is a natural generalization of traditional local edge-coloring. Our main results include sufficient conditions on the function $f$ to construct such colorings. As corollaries, we obtain local analogs of Vizing and Shannon's theorems, recovering a recent result of Conley, Grebík and Pikhurko. |
| title | Multigraph edge-coloring with local list sizes |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2307.12094 |