Cyclically Interval Total Colorings of Cycles and Middle Graphs of Cycles

A total coloring of a graph G is a function ( ) ( ) : E G V G α →   such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. A k -interval is a set of k consecutive integers. A cyclically interval total t -coloring of a graph G is a total coloring α of G with colors 1,2, , t  such that at least one vertex or edge of G is colored by , 1, 2, , i i t =  , and for any ( ) x V G ∈ , the set [ ] ( ) { } ( ) { } , is incident to S v v e e v α α α =  is a ( ) ( ) 1 G d x + -interval, or { } [ ] 1,2, , \ , t S x α  is a ( ) ( ) 1 G t d x − − -interval, where ( ) G d x is the degree of the vertex x in G. In this paper, we study the cyclically interval total colorings of cycles and middle graphs of cycles.

t -coloring of a graph G is a total coloring α of G with colors 1, 2, ,t  such that at least one vertex or edge of G is colored by , 1, 2, , i i t =  , and for any ( )

Introduction
All graphs considered in this paper are finite undirected simple graphs.For a graph G, let ( ) V G and ( ) E G denote the set of vertices and edges of G, respectively.For a vertex ( ) x denote the degree of x in G.
We denote ( ) the maximum degree of vertices of G.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 [ ] A total coloring of a graph G is a function ( ) ( )  such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color.
The concept of total coloring was introduced by Vizing [1] and independently by Behzad [2].The total chromatic number ( ) G χ′′ is the smallest number of colors needed for total coloring of G.For a total coloring α of a graph G and for any An interval total t -coloring of a graph G is a total coloring of G with colors 1, 2, ,t  such that at least one vertex or edge of G is colored by , 1, 2, , i i t =  , and for any ( ) x α is a ( ) ( ) interval total colorable if it has an interval total t -coloring for some positive integer t.
For any t ∈  , let t T denote the set of graphs which have an interval total t -coloring, and let Clearly, The concept of interval total coloring was first introduced by Petrosyan [3].
Now we generalize the concept interval total coloring to the cyclically interval total coloring.A total t -coloring α of a graph G is called a cyclically interval total t -coloring of G, if for any ( ) ) total colorable if it has a cyclically interval total t -coloring for some positive integer t.For any t ∈  , we denote by t F the set of graphs for which there exists a cyclically interval total t -coloring.Let . It is also clear that for any G ∈ T , the following inequality is true and in which two vertices are adjacent whenever either they are adjacent edges of G or one is a vertex of G and other is an edge incident with it.
In this paper, we study the cyclically interval total colorings of cycles and middle graphs of cycles.For a cycle n C , let , , , , , , , where 1 1 n e v v = and  .For example, the graphs in Figure 1

Cn
In this section we study the cyclically interval total colorings of ( ) .In [4] it was proved the following result.Theorem 1. (H.P. Yap [4]) For any integer Proof.Since ⊆ T F , then for any G ∈ T we have [ ] [ ] [ ] [ ] Proof.By contradiction.Suppose that, for any integers 4 n ≥ , α is a cyclically interval total ( ) Without loss of generality, we may assume that ( ) ( ) , there is only one vertex or one edge of n C is colored by k .Case 1.At least one of i and j is even.Say that i is even.Without loss of generality, suppose that 2 i n = , i.e., ( ) without loss of generality, we may assume that − .On the other hand, since α is a cyclically interval total ( ) weather j is odd or even.A contradiction.Case 2. i and j are all odd.Without loss of generality, suppose that 1 i = .Then we have 3 Note that i a and j a are all vertices of n C .Since α is a cyclically interval total ( ) without loss of generality, we may assume that

n M C
In this section we study the cyclically interval total colorings of ( ) ( ) Case 1. n is even.
By the definition of α we have , , .
n e e α = See Figure 3.
By the definition of α we have   Proof.Now we define a total 4n-coloring α of the graph ( )     ( ) Then we have ( ) 1 n e e + = .See Figure 5.
By the definition of α we have , e e + = .See Figure 6.
By the definition of α we have ( ) . Then we have ( ) , , e e + = .See Figure 7.
By the definition of α we have 1 n e e + = .See Figure 8.
By the definition of α we have 1 n e e + = .See Figure 10.
By the definition of α we have the sets of even and odd integers in [ ] , a b , respectively.An interval D is called a h -interval if D h = .
For a graph G ∈ T , the least and the greatest values of t for which t G ∈ T are denoted by

are 4 C and ( ) 4 M
C , respectively.Note that in Section 3 we always use the kind of diagram like (c) in Figure 1 to denote ( ) n M C .
Figure 1. 4 C and i

3 1
we consider the color of 3 a .By the definition of α , ( ) {} not be 1, 3 or 2 1 n −obviously.So we have ( ) is a contradiction to that integer 3 n ≥ .Now we define a total 5-coloring α of the graph ( ) n M C as follows:

4 .
By the definition of α we have

[
This shows that α is a cyclically interval total 4n-coloring of

.
We define a total k -coloring α of ( ) n M C as follows.First we use the colors 1, 2, , k  to color the vertices and edges of ( ) n M C beginning from 1 e by the way used in the proof of Theorem 9. Now we color the other vertices and edges of ( ) n M C with the colors 1, 2, , k  .

(
See Figure9.By the definition of α we have