The planar edge-coloring theorem of Vizing in $O(n\log n)$ time
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| Format: | Preprint |
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2025
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| _version_ | 1866915980239699968 |
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| author | Jędrzejczak, Patryk Kowalik, Łukasz |
| author_facet | Jędrzejczak, Patryk Kowalik, Łukasz |
| contents | In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $Δ\ge 8$ can be edge-colored using $Δ$ colors. The direct implementation of the Vizing's proof gives an algorithm that finds the coloring in $O(n^2)$ time for an $n$-vertex input graph. Chrobak and Nishizeki [J. Algorithms, 1990] have shown a more careful algorithm, which improves the time to $O(n\log n)$ time, though only for $Δ\ge 9$. In this paper, we extend their ideas to get an algorithm also for the missing case $Δ=8$. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The planar edge-coloring theorem of Vizing in $O(n\log n)$ time Jędrzejczak, Patryk Kowalik, Łukasz Data Structures and Algorithms G.2.2 In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $Δ\ge 8$ can be edge-colored using $Δ$ colors. The direct implementation of the Vizing's proof gives an algorithm that finds the coloring in $O(n^2)$ time for an $n$-vertex input graph. Chrobak and Nishizeki [J. Algorithms, 1990] have shown a more careful algorithm, which improves the time to $O(n\log n)$ time, though only for $Δ\ge 9$. In this paper, we extend their ideas to get an algorithm also for the missing case $Δ=8$. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs. |
| title | The planar edge-coloring theorem of Vizing in $O(n\log n)$ time |
| topic | Data Structures and Algorithms G.2.2 |
| url | https://arxiv.org/abs/2507.04516 |