Cyclically Interval Total Coloring of the One Point Union of Cycles

A total coloring of a graph G with colors 1,2, , t  is called a cyclically interval total t-coloring if all colors are used, and the edges incident to each vertex ( ) v V G ∈ together with v are colored by ( ) ( ) 1 G d v + consecutive colors modulo t, where ( ) G d v is the degree of the vertex v in G. The one point union ( ) k n C of k-copies of cycle n C is the graph obtained by taking v as a common vertex such that any two distinct cycles n C′ and n C′′ are edge disjoint and do not have any vertex in common except v. In this paper, we study the cyclically interval total colorings of ( ) k n C , where 3 n ≥ and 2 k ≥ .


C
of k-copies of cycle n C is the graph obtained by taking v as a

Introduction
We denote the sets of vertices and edges in a graph G by ( ) V G and ( ) E G , respectively.For a vertex ( ) , we denote the degree of x in G by ( ) and we use ( ) to denote the maximum degree of vertices of G.For an arbitrary finite set A, we denote the number of elements of A by A .
We use  to denote the set of positive integers.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 = .
A total coloring of a graph G is a function mapping ( ) ( ) such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color.The concept of total coloring was introduced by V. Vizing [1] and independently by M. 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, ,  , and for any ( ) ) A graph G is interval total colorable if it has an interval total t-coloring for some positive integer t.The concept of interval total coloring was first introduced by Petrosyan [3].
Recently, Zhao and Su [4] generalized the concept interval total coloring to the cyclically interval total coloring as follow.A total t-coloring α of a graph G is called a cyclically interval total t-coloring of G, if for any ( ) ) G is cyclically interval 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 clear that for any G ∈ F , the following inequality is true: The one point union ( )   k n

C
of k-copies of cycle n C is the graph obtained by taking v as a common vertex such that any two distinct cycles n C′ and n C′′ are edge disjoint and do not have any vertex in common except v.In this paper, we study the cyclically interval total colorability of ( ) , , , Without loss of generality, we may assume that the common vertex v of the k-copies of cycle n C is the first vertex in each cycle, i.e., For example, the graphs in Figure 1 are all ( ) C .Note that in the paper we al- ways use the kind of diagram like (b) in Figure 1 to denote ( ) All graphs considered in this paper are finite undirected simple graphs.

Main Results
Vaidya and Isaac [5] studied the total coloring of ( ) and got the following result.
Theorem 1 (Vaidya and Isaac) For any integers Now we consider the cyclically interval total colorings of ( ) Theorem 2 For any integers , , , . Without loss of generality, we may assume that the common vertex v of the k-copies of cycle n C is the first vertex in each cycle, i.e., as follows: Case 1.
By the definition of α, we have , and 1, .
This shows that α is a cyclically interval total ( ) ( ) , , By the definition of α, we have ; This shows that α is a cyclically interval total ( ) ( ) , , , , 4 for Open Journal of Discrete Mathematics  By the definition of α, we have 1 and 1, ; This shows that α is a cyclically interval total ( ) Theorem 3 For any integers Proof.Suppose that 3 n ≥ and 2 k ≥ .We consider the following two cases.Case 1.
2 2 k n ≤ − .Now we define a total ( ) ; , . See Figure 5 for an example.Open Journal of Discrete Mathematics By the definition of α, we have .
This shows that α is a cyclically interval total ( ) as follows: Let ( ) . See Figure 6 for an example.
By the definition of α, we have 2, and 1, .
This shows that α is a cyclically interval total ( ) C .

Generalization
The one point of union ( ) k C of any k cycles By the proof of Theorem 2, the following definitions are well defined.
Definition 4 A partial ( ) Now we consider the cyclically interval total colorings of ( ) k C .
Theorem 5 For any integer is the one point of union of cycles , , , . Without loss of generality, we may assume that the common vertex v of the k cycles the degree of the vertex v in G.The one point union ( ) k n For any graph G ∈ F , the minimum and the maximum values of t for which G has a cyclically interval total t-coloring are denoted by

,
and provide a lower bound of Open Journal of Discrete Mathematics

Figure 3
Figure 3 for an example.

C
obtained by taking v as a common vertex such that any two distinct cycles are edge disjoint and do not have any vertex in common except v.
first vertex in each cycle, i.e.,

C 1 k
, respectively.By the definition of α, Open Journal of Discrete Mathematics 2 + is the largest color used in coloring α, and [ ] [ ] , we consider the one point of union( )   further research maybe more interesting.