It was not known about simple cycles

TL;DR

This paper characterizes the possible numbers of colors for proper edge colorings of simple cycles where adjacent edges are colored with consecutive or wrap-around colors, expanding understanding of cycle colorings.

Contribution

It determines all feasible color counts for proper edge colorings of simple cycles with specific adjacency color constraints.

Findings

Identifies all t for which such colorings exist for C(n)

Provides a complete characterization of these colorings for all n≥3

Enhances understanding of cycle edge coloring constraints

Abstract

A proper edge $t$-coloring of a graph is a coloring of its edges with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. For any integer $n\geq 3$, all possible values of $t$ are found, for which there exists such a proper edge $t$-coloring of the simple cycle C(n), which uses for each pair of adjacent edges either consecutive colors or the first and the last ones.