On the zero-free region for the chromatic polynomial of graphs with maximum degree $Δ$ and girth $g$
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_ | 1866909321949872128 |
|---|---|
| author | Fialho, Paula M. S. Juliano, Emanuel Procacci, Aldo |
| author_facet | Fialho, Paula M. S. Juliano, Emanuel Procacci, Aldo |
| contents | The purpose of the present paper is to provide, for all pairs of integers $(Δ,g)$ with $\D\ge 3$ and $g\ge 3$, a positive number $C(Δ, g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $Δ$ and finite girth $g$ is free of zero if $|q|\ge C(Δ, g)$. Our bounds enlarge the zero-free region in the complex plane of $P_G(q)$ in comparison to previous bounds. In particular, for small values of $\D$ our estimates yield a sensible improvement on the bounds recently obtained by Jenssen, Patel and Regts in \cite{JPR}, while they coincide with those of \cite{JPR} when $Δ\to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13892 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the zero-free region for the chromatic polynomial of graphs with maximum degree $Δ$ and girth $g$ Fialho, Paula M. S. Juliano, Emanuel Procacci, Aldo Combinatorics 05C31 The purpose of the present paper is to provide, for all pairs of integers $(Δ,g)$ with $\D\ge 3$ and $g\ge 3$, a positive number $C(Δ, g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $Δ$ and finite girth $g$ is free of zero if $|q|\ge C(Δ, g)$. Our bounds enlarge the zero-free region in the complex plane of $P_G(q)$ in comparison to previous bounds. In particular, for small values of $\D$ our estimates yield a sensible improvement on the bounds recently obtained by Jenssen, Patel and Regts in \cite{JPR}, while they coincide with those of \cite{JPR} when $Δ\to \infty$. |
| title | On the zero-free region for the chromatic polynomial of graphs with maximum degree $Δ$ and girth $g$ |
| topic | Combinatorics 05C31 |
| url | https://arxiv.org/abs/2409.13892 |