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A Note on ID-Colorings and Symmetric Colorings of Cycles

Yuya Kono

TL;DR

The paper proves that for a cycle $C_n$ with prime order $n$, a red-white coloring is an ID-coloring precisely when it is not symmetric about any vertex, using a symmetry-based framework that parallels recent path results. It introduces central-vertex concepts, an explicit algorithm to extract symmetry from non-ID colorings, and a rigorous justification of termination via a sequence of partner-pairs. The work also shows that the prime-order condition is essential by constructing colorings on non-prime cycles that are neither ID-colorings nor symmetric, while odd non-prime cycles admit symmetric colorings with multiple central vertices. Collectively, the results provide a concrete criterion for ID-colorings on prime cycles and offer tools potentially applicable to broader graph classes containing cycles as subgraphs.

Abstract

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.

A Note on ID-Colorings and Symmetric Colorings of Cycles

TL;DR

The paper proves that for a cycle with prime order , a red-white coloring is an ID-coloring precisely when it is not symmetric about any vertex, using a symmetry-based framework that parallels recent path results. It introduces central-vertex concepts, an explicit algorithm to extract symmetry from non-ID colorings, and a rigorous justification of termination via a sequence of partner-pairs. The work also shows that the prime-order condition is essential by constructing colorings on non-prime cycles that are neither ID-colorings nor symmetric, while odd non-prime cycles admit symmetric colorings with multiple central vertices. Collectively, the results provide a concrete criterion for ID-colorings on prime cycles and offer tools potentially applicable to broader graph classes containing cycles as subgraphs.

Abstract

A red-white coloring of a nontrivial connected graph is an assignment of red and white colors to the vertices of~. Associated with each vertex of of diameter is a -vector, called the code of , whose th coordinate is the number of red vertices at distance from . A red-white coloring of for which distinct vertices have distinct codes is called an ID-coloring of . 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.

Paper Structure

This paper contains 13 sections, 24 theorems, 6 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Main Theorem
  3. Symmetric Colorings of Odd Cycles
  4. The Statement of the Main Theorem
  5. Tools and Terminologies
  6. An Algorithm and an Example
  7. How the Algorithm Works
  8. The Converse of Theorem \ref{['prime-id-pw1']}
  9. Observations for Cycles of Non-Prime Order
  10. Cycles of Non-Prime Odd Order
  11. Cycles of Even Order
  12. Codes of Symmetric Colorings of Cycles
  13. Remarks

Key Result

Theorem 1.1

Let $n\ge 2$ and let $c$ be a red-white coloring of the path $P_n$ under which the end vertices of $P_n$ are colored red. The coloring $c$ is an ID-coloring if and only if $c$ is not symmetric.

Figures (6)

  • Figure 1: Symmetric colorings of $P_6$ and $P_9$
  • Figure 2: A symmetric coloring of $C_{13}$ with respect to the vertex $u$
  • Figure 3: All possible red-white colorings of $C_3$ and $C_{5}$
  • Figure 4: The SA-colorings of $C_9$, $C_{15}$ and $C_{25}$ with 3, 3, and 5 splitting vertices
  • Figure 5: A symmetric coloring of $C_{10}$
  • ...and 1 more figures

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.7
  • ...and 14 more