Borel Vizing's Theorem for Graphs of Subexponential Growth
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909291866226688 |
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| author | Bernshteyn, Anton Dhawan, Abhishek |
| author_facet | Bernshteyn, Anton Dhawan, Abhishek |
| contents | We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $Δ(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $Δ(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00095 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Borel Vizing's Theorem for Graphs of Subexponential Growth Bernshteyn, Anton Dhawan, Abhishek Combinatorics Distributed, Parallel, and Cluster Computing Logic We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $Δ(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $Δ(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$. |
| title | Borel Vizing's Theorem for Graphs of Subexponential Growth |
| topic | Combinatorics Distributed, Parallel, and Cluster Computing Logic |
| url | https://arxiv.org/abs/2307.00095 |