Chromatic Polynomial Evaluation Spectra
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909973709062144 |
|---|---|
| author | Miyazaki, Rafael Pohoata, Cosmin Zheng, Michael |
| author_facet | Miyazaki, Rafael Pohoata, Cosmin Zheng, Michael |
| contents | Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial $F_{G}\left(\frac{3+\sqrt{5}}{2}\right)$ already takes on exponentially many values, if one varies $G$ over all planar cubic graphs $G$ on $n$ vertices. We show, more generally, that the size of the set $\{P_G(q): |V(G)|=n\}$ is exponential in $n$, for every fixed real number $q \neq 0,1,2$. In fact, our approach can also be pushed to show that $P_{G}(q)$ already takes on exponentially many values, if we only vary $G$ over all planar graphs on $n$ vertices. The case $q=3$ confirms a conjecture of Agol, which was initially motivated by the $\mathsf{NP}$-completeness of planar $3$-colorability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19600 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chromatic Polynomial Evaluation Spectra Miyazaki, Rafael Pohoata, Cosmin Zheng, Michael Combinatorics Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial $F_{G}\left(\frac{3+\sqrt{5}}{2}\right)$ already takes on exponentially many values, if one varies $G$ over all planar cubic graphs $G$ on $n$ vertices. We show, more generally, that the size of the set $\{P_G(q): |V(G)|=n\}$ is exponential in $n$, for every fixed real number $q \neq 0,1,2$. In fact, our approach can also be pushed to show that $P_{G}(q)$ already takes on exponentially many values, if we only vary $G$ over all planar graphs on $n$ vertices. The case $q=3$ confirms a conjecture of Agol, which was initially motivated by the $\mathsf{NP}$-completeness of planar $3$-colorability. |
| title | Chromatic Polynomial Evaluation Spectra |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.19600 |