Extremal diameters of 3-coloring graphs of trees
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908746194616320 |
|---|---|
| author | Asgarli, Shamil Krehbiel, Sara MacLean, Simon Zaimi, Gjergji |
| author_facet | Asgarli, Shamil Krehbiel, Sara MacLean, Simon Zaimi, Gjergji |
| contents | Given a tree $T$, its 3-coloring graph $\mathcal{C}_3(T)$ has as vertices the proper 3-colorings of $T$, with edges joining colorings that differ at exactly one vertex. We call the diameter of $\mathcal{C}_3(T)$ the 3-coloring diameter of $T$. We introduce the notion of balanced labelings of $T$ and show that the 3-coloring diameter equals the maximum $L_1$-norm of a balanced labeling. Using this equivalence, we determine the maximum and minimum values of the 3-coloring diameter over all trees on $n$ vertices and characterize the extremal trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal diameters of 3-coloring graphs of trees Asgarli, Shamil Krehbiel, Sara MacLean, Simon Zaimi, Gjergji Combinatorics Primary 05C15, Secondary 05C05, 05C12, 05C35 Given a tree $T$, its 3-coloring graph $\mathcal{C}_3(T)$ has as vertices the proper 3-colorings of $T$, with edges joining colorings that differ at exactly one vertex. We call the diameter of $\mathcal{C}_3(T)$ the 3-coloring diameter of $T$. We introduce the notion of balanced labelings of $T$ and show that the 3-coloring diameter equals the maximum $L_1$-norm of a balanced labeling. Using this equivalence, we determine the maximum and minimum values of the 3-coloring diameter over all trees on $n$ vertices and characterize the extremal trees. |
| title | Extremal diameters of 3-coloring graphs of trees |
| topic | Combinatorics Primary 05C15, Secondary 05C05, 05C12, 05C35 |
| url | https://arxiv.org/abs/2512.03789 |