Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929357495205888 |
|---|---|
| author | Bhattacharya, Sayan Carmon, Din Costa, Martín Solomon, Shay Zhang, Tianyi |
| author_facet | Bhattacharya, Sayan Carmon, Din Costa, Martín Solomon, Shay Zhang, Tianyi |
| contents | Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985].
In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15449 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier Bhattacharya, Sayan Carmon, Din Costa, Martín Solomon, Shay Zhang, Tianyi Data Structures and Algorithms Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985]. In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years. |
| title | Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2405.15449 |