It was not known about simple cycles

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 ≥ 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. We consider undirected, simple, finite and connected graphs. For a graph G, we denote by V (G) and E(G) the sets of its vertices and edges, respectively. The set of edges of G incident with a vertex x ∈ V (G) is denoted by J G (x). For any x ∈ V (G), d G (x) denotes the degree of the vertex x in G. For a graph G, ∆(G) denotes the maximum degree of a vertex of G. A simple cycle with n edges (n ≥ 3) is denoted by C(n). A simple path with n edges (n ≥ 1) is denoted by P (n). For an arbitrary finite set A, we denote by |A| the number of elements of A. The set of positive integers is denoted by N. For any subset D of the set N, we denote by D (0) and D (1) the subsets of all even and all odd elements of D, respectively. An arbitrary nonempty subset of consecutive integers is called an interval. An interval with the minimum element p and the maximum element q is denoted by [p, q]. An interval D is called a h-interval if |D| = h. For any positive integer k define ε(k) ≡ 1 + ⌊ k 2 ⌋ − ⌈ k 2 ⌉. For any nonnegative integer k define sgn(k) ≡ 0, if k = 0 1 otherwise. A function ϕ : E(G) → [1, t] is called a proper edge t-coloring of a graph G, if all colors are used, and no two adjacent edges receive the same color. The minimum value of t for which there exists a proper edge t-coloring of a graph G is denoted by χ ′ (G) [10]. If G is a graph, and ϕ is its proper edge t-coloring, where t ∈ [χ ′ (G), |E(G)|], then we define U (G, ϕ) ≡ {e ∈ E(G)/1 < ϕ(e) < t}. …