Note on Cyclically Interval Edge Colorings of Simple Cycles

A proper edge t-coloring of a graph G is a coloring of its edges with colors t 1, 2, , such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex ( ) x V G ∈ , either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.


Introduction
All graphs considered in this paper are finite undirected simple graphs.For a graph G, let ( ) V G and ( ) E G denote the sets of vertices and edges of G, respectively.For a vertex ( ) ( ) G d x denote the subset of ( ) E G incident with the vertex x, and the degree of the vertex x in G, respectively.We denote ( ) the maximum degree of vertices of G.A simple path with 1 n ≥ edges is denoted by n P .A simple cycle with 3 n ≥ edges is denoted by n C .For an arbitrary finite set A, we denote by A the number of elements of A. The set of positive integers is denoted by  .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 .We denote [ ] 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 exists a proper edge t-coloring of a graph G is denoted by ( ) A graph G is interval colorable if it has an interval t-coloring for some positive integer t.The concept of interval edge coloring of graphs was introduced by Asratian and Kamalian [1].In [1] [2], the authors showed that if G is interval colorable, then . For any t ∈  , we denote by t N the set of graphs for which there exists an interval t-coloring.Let , at least one of the following two conditions holds: 1) A graph G is interval cyclically colorable if it has a cyclically interval t-coloring for some positive integer t.This type of edge coloring under the name of "π-coloring" was first considered by Kotzig [3], and the concept of cyclically interval edge coloring of graphs was explicitly introduced by de Werra and Solot [4].
For any t ∈  , we denote by t M the set of graphs for which there exists a interval cyclic t-coloring.Let M and ⊆ N M. Note that for an arbitrary graph G, ( ) ( ) . It is also clear that for any G ∈ N , the following inequality is true: Let T be a tree.Kamalian [5] [6] showed that T ∈ N , ( ) ( ) W T .Kamalian [7] [8] also proved that ( ) ( ) Some interesting results on cyclically interval t-colorings and related topics were obtained in [3] [4] [9]- [14].For any integer 3 n ≥ , Kamalian [13] proved that n C ∈ M , determined the set , and provided the following theorem.Theorem 1 (R. R. Kamalian [13]) For any integers In this paper, we provide a new proof of the theorem.The terms and concepts that we do not define can be found in [15].

Main Result
Proof of Theorem 1. Suppose that, in clockwise order along the cycle n C , the vertices of n C are 1 2 , , , n v v v and the edges of n C are 1 2 , , , n e e e , where , and . .
( ) E F is odd, and then ( ) ( ) H is a graph with m connected components, 2 m ≥ .Suppose that, in clockwise order along the cycle n C , the m connected components of 0 H are 1 2 , , , m H H H . Without loss of generality, we may also assume that ( ) sets of even and odd integers in [ ]

,
and α is a proper edge t-coloring of a graph G, edge t-coloring α of a graph G is called an interval t-coloring of G if for any any graph G ∈ N , the minimum and the maximum values of t for which G has an interval t-coloring are denoted by ( ) w G and any graph G ∈ M , the minimum and the maximum values of t for which G has a cyclically interval t-coloring are denoted by

θ
was an interval, and provided the exact values of the parameters ( ) w T and

3 L
be the subgraph induced by { }