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A proper edge
*t*-coloring of a graph G 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. A cyclically interval
*t*-coloring of a graph G is a proper edge
*t*-coloring of G such that for each vertex
, 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”.

All graphs considered in this paper are finite undirected simple graphs. For a graph G, let

For an arbitrary finite set A, we denote by

A function

For any

t-coloring are denoted by

A proper edge t-coloring a of a graph G is called a interval cyclic t-coloring of G, if for any

1)

2)

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 “p-coloring” was first considered by Kotzig [

For any

interval t-coloring are denoted by

It is clear that for any

Let T be a tree. Kamalian [

Theorem 1 (R. R. Kamalian [

In this paper, we provide a new proof of the theorem. The terms and concepts that we do not define can be found in [

Proof of Theorem 1. Suppose that, in clockwise order along the cycle

We know that if

then

First we prove that if

then

Case 1. n is odd.

For any

It is easy to check that

Case 2. n is even.

For any

If

For any

It is easy to check that, in each case,

Now let us prove that if

By contradiction. Suppose that there are

such that

Case 1.

Clearly,

be the subgraph induced by

Case 2.

Let H be the graph removing from the graph

Case 2.1.

Let F be the subgraph of

Clearly,

coloring of

If

Case 2.2.

Suppose that, in clockwise order along the cycle

Clearly,

and L_{4} be the subgraph induced by

say

coloring with

Now let

tradiction. □

We thank the editor and the referee for their valuable comments. Research of Y. Zhao is funded in part by the Natural Science Foundation of Hebei Province of China under Grant No. A2015106045, and in part by the Institute of Applied Mathematics of Shijiazhuang University.

Nannan Wang,Yongqiang Zhao, (2016) Note on Cyclically Interval Edge Colorings of Simple Cycles. Open Journal of Discrete Mathematics,06,180-184. doi: 10.4236/ojdm.2016.63016