Almost colour-balanced spanning forests in complete graphs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910641121394688 |
|---|---|
| author | Hollom, Lawrence Mond, Adva Portier, Julien |
| author_facet | Hollom, Lawrence Mond, Adva Portier, Julien |
| contents | Given $K_n$ whose edges are coloured red and blue, and a forest $F$ of order $n$, we seek embeddings of $F$ with small imbalance, that is, difference between the numbers of red and blue edges. We show that if the $2$-colouring of the edges of $K_n$ is balanced, meaning that the numbers of red and blue edges are equal, and $F$ has maximum degree $Δ$, then one can find an embedding of $F$ into $K_n$ whose imbalance is at most $Δ/2 + 18$, which is essentially best possible and resolves a conjecture of Mohr, Pardey, and Rautenbach. Furthermore, we give a tighter bound for the imbalance for small values of $Δ$. In particular, we prove that the imbalance can be taken to be constant in the case where $Δ<n(1/4 - η)$ for any constant $η>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06148 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost colour-balanced spanning forests in complete graphs Hollom, Lawrence Mond, Adva Portier, Julien Combinatorics Given $K_n$ whose edges are coloured red and blue, and a forest $F$ of order $n$, we seek embeddings of $F$ with small imbalance, that is, difference between the numbers of red and blue edges. We show that if the $2$-colouring of the edges of $K_n$ is balanced, meaning that the numbers of red and blue edges are equal, and $F$ has maximum degree $Δ$, then one can find an embedding of $F$ into $K_n$ whose imbalance is at most $Δ/2 + 18$, which is essentially best possible and resolves a conjecture of Mohr, Pardey, and Rautenbach. Furthermore, we give a tighter bound for the imbalance for small values of $Δ$. In particular, we prove that the imbalance can be taken to be constant in the case where $Δ<n(1/4 - η)$ for any constant $η>0$. |
| title | Almost colour-balanced spanning forests in complete graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.06148 |