Simultaneous edge-colourings
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912107545493504 |
|---|---|
| author | Boyadzhiyska, Simona Lang, Richard Lo, Allan Molloy, Michael |
| author_facet | Boyadzhiyska, Simona Lang, Richard Lo, Allan Molloy, Michael |
| contents | We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the maximum degree $Δ$, for small values of $k$ and for an infinite sequence of values of $k$. This asymptotically settles a conjecture of Cabello for $k=2$. Moreover, we show that $\sqrt k Δ+ o(Δ)$ colours always suffice, which tends to the optimal value as $k$ grows. We also show that $\ell Δ+ o(Δ)$ colours are enough when every edge appears in at most $\ell$ of the graphs, which asymptotically confirms a conjecture of Cambie. Finally, our results extend to the list setting. We also find a close connection to a conjecture of Füredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simultaneous edge-colourings Boyadzhiyska, Simona Lang, Richard Lo, Allan Molloy, Michael Combinatorics 05C15, 05D15, 05D40 G.2.2 We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the maximum degree $Δ$, for small values of $k$ and for an infinite sequence of values of $k$. This asymptotically settles a conjecture of Cabello for $k=2$. Moreover, we show that $\sqrt k Δ+ o(Δ)$ colours always suffice, which tends to the optimal value as $k$ grows. We also show that $\ell Δ+ o(Δ)$ colours are enough when every edge appears in at most $\ell$ of the graphs, which asymptotically confirms a conjecture of Cambie. Finally, our results extend to the list setting. We also find a close connection to a conjecture of Füredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings. |
| title | Simultaneous edge-colourings |
| topic | Combinatorics 05C15, 05D15, 05D40 G.2.2 |
| url | https://arxiv.org/abs/2411.04071 |