A Note on ID-Colorings and Symmetric Colorings of Cycles
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912314200948736 |
|---|---|
| author | Kono, Yuya |
| author_facet | Kono, Yuya |
| contents | A red-white coloring of a nontrivial connected graph $G$ is an assignment of red and white colors to the vertices of~$G$. Associated with each vertex $v$ of $G$ of diameter $d$ is a $d$-vector, called the code of $v$, whose $i$th coordinate is the number of red vertices at distance $i$ from $v$. A red-white coloring of $G$ for which distinct vertices have distinct codes is called an ID-coloring of $G$. In 2025, a criterion to determine whether a red-white coloring of a path is an ID-coloring or not was presented by Kono, with the aid of a result shown by Marcelo et al. in 2024. The criterion utilizes the fact that ID-colorings of paths are ``opposite'' of colorings with a certain symmetry. In this paper, we establish a similar criterion that can be applied for cycles whose order is a prime number at least 3. In order to do so, we employ an analogous approaches used for the criterion for paths, i.e., we pay attention to symmetries of given red-white colorings of cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on ID-Colorings and Symmetric Colorings of Cycles Kono, Yuya Combinatorics A red-white coloring of a nontrivial connected graph $G$ is an assignment of red and white colors to the vertices of~$G$. Associated with each vertex $v$ of $G$ of diameter $d$ is a $d$-vector, called the code of $v$, whose $i$th coordinate is the number of red vertices at distance $i$ from $v$. A red-white coloring of $G$ for which distinct vertices have distinct codes is called an ID-coloring of $G$. In 2025, a criterion to determine whether a red-white coloring of a path is an ID-coloring or not was presented by Kono, with the aid of a result shown by Marcelo et al. in 2024. The criterion utilizes the fact that ID-colorings of paths are ``opposite'' of colorings with a certain symmetry. In this paper, we establish a similar criterion that can be applied for cycles whose order is a prime number at least 3. In order to do so, we employ an analogous approaches used for the criterion for paths, i.e., we pay attention to symmetries of given red-white colorings of cycles. |
| title | A Note on ID-Colorings and Symmetric Colorings of Cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.05340 |