On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles

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 its vertex x, 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. For an arbitrary simple cycle, all possible values of t are found, for which the graph has a cyclically interval t-coloring.


Introduction
We consider undirected, simple, finite and connected graphs.For a graph , we denote by and the sets of its vertices and edges, respectively.The set of edges of incident with a vertex .The terms and concepts that we do not define can be found in [1].

  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 .For any subset of the set , we denote by   0 and   1 the subsets of all even and all odd elements of , respectively.
An arbitrary nonempty subset of consecutive integers is called an interval.An interval with the minimum element and the maximum element is denoted by For any real number  , we denote by             the maximum (minimum) integer which is less (greater) than or equal to  .
For any positive integer define k For any nonnegative integer define

   
A function :  1 , is called a proper edge -coloring of a graph G, if all colors are used, and no two adjacent edges receive the same color.
The minimum value of for which there exists a proper edge t -coloring of a graph is denoted by . If is a graph, and  is its proper edge -coloring, where For any , we denote by t the set of graphs for which there exists an interval -coloring.Let , at least one of the following two nditions ho 1) . It is also clear that for any G  N , e followi ality is true: r any tree G hat G In [5,6]
In this paper, for any integer Proposition 1.For any integer 3 for which a cyclically interval 0 t -coloring Let us construct a graph 0 H removing from the graph 00 H all its isolated vertice Case A. 0 s.H is a connected graph.o Let us den te by F the simple path with pendant edges e and e which is isomorphic to the graph . It means that 0 t is an even number, satisfying the inequality Clearly, , we c clude from the definition f 0

C
on o H , that fo ph r a gra F , there exists an interval bered in succession at bypassing mentioned in 1), 4)   For any , we define a point π j of the 2- dimensional rectangle coordinate system by e following way: th .
Case B.2. 0 n is even.Clearly, is represented as a sum of two even numb It means that quality ot difficult to see that in this case, for an arbitrary is odd, and, oreover, for ently, there are di fferent integers i In this case, evid  and i in the set   Without loss of generality assume that  is represented as a sum of one odd and two even numbers, which is impossible.

 
Clearly, for any  contains exactly one of the colors 1 and 0 0  .It means that there exists an interval   Let   It is not difficult to see that there exists an interval oring of the graph H  which colors pendant edges of Consequently, we obtain that Copyright © 2013 SciRes.OJDM which is impossible.Thus, we have pr n   , 5 n  and oved that if Clearly, there exists 0 m   , to note that the existence of an interval 0 t -coloring of a graph Now, to see a contradiction, it is enough with the existence of an interval 2-coloring of a graph rem 1 is proved.It means that we also have Theorem 2. For an arbitrary integer 5 n  ,

Acknowledgements
The author thanks P.A. Petrosyan and N.A. Khachatryan for their attention to this work.
is not difficult to see that there exists an interval H  hich colors pendant edge 3 i w  s of H by the colo 1.Hence, r have proved, that if n   , we denote by int and  For any   respectively, for which t G int W G the minimum and the maximum possible value of t , N .For a graph G , let us  we can conclude from the the graph H  is called horizontal if the points π  and π  have the same ordinate.
  | e both even. and,