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http://www.scirp.org/journal/ojdm ISSN Online: 2161-7643

ISSN Print: 2161-7635

DOI: 10.4236/ojdm.2017.74018 Oct. 31, 2017 200 Open Journal of Discrete Mathematics

Cyclically Interval Total Colorings of Cycles

and Middle Graphs of Cycles

Yongqiang Zhao

1

, Shijun Su

2

1School of Science, Shijiazhuang University, Shijiazhuang, China 2School of Science, Hebei University of Technology, Tianjin, China

Abstract

A total coloring of a graph G is a function

α

:

E G

( )

V G

( )

such that no

adjacent vertices, edges, and no incident vertices and edges obtain the same col-or. 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 i i, =1, 2,,t, and for any

( )

x V G

, the set S

[ ]

α,v =

{

α

( )

v

}

{

α

( )

e eis incident tov

}

is a

( )

(

dG x +1

)

-interval, or

{

1, 2,

,

t

}

\

S

[ ]

α

,

x

is a

(

tdG

( )

x −1

)

-interval,

where

d

G

( )

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.

Keywords

Total Coloring, Interval Total Coloring, Cyclically Interval Total Coloring, Cycle, Middle Graph

1. 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 V G

( )

, let

d

G

( )

x

denote the degree of x in G.

We denote

( )

G

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 b

,

and

[ ]

a b

,

the sets of even and odd integers in

[ ]

a b

,

, respectively. An interval D is called a

h

-interval if

D

=

h

.

How to cite this paper: Zhao, Y.Q. and Su, S.J. (2017) Cyclically Interval Total Color-ings of Cycles and Middle Graphs of Cycles. Open Journal of Discrete Mathe-matics, 7, 200-217.

https://doi.org/10.4236/ojdm.2017.74018

Received: August 10, 2017 Accepted: October 28, 2017 Published: October 31, 2017

Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

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DOI: 10.4236/ojdm.2017.74018 201 Open Journal of Discrete Mathematics

A total coloring of a graph G is a function

α

:

E G

( )

V G

( )

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

χ

′′

( )

G

is the smallest number of

colors needed for total coloring of G. For a total coloring α of a graph G and for any

v V G

( )

, let S

[ ]

α,v =

{

α

( )

v

}

{

α

( )

e eis incident tov

}

.

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 i i, =1, 2,,t, and for any

x V G

( )

, the set

S

[ ]

α

,

x

is a

(

dG

( )

x +1

)

-interval. A graph G is

interval total colorable if it has an interval total t-coloring for some positive integer t.

For any t∈, let Tt denote the set of graphs which have an interval total t -coloring, and let

T

=

t≥1

T

t. For a graph G∈T, the least and the greatest

values of t for which G∈Tt are denoted by

w G

τ

( )

and

W G

τ

( )

, respectively.

Clearly,

( )

G w Gτ

( )

W Gτ

( )

V G

( )

E G

( )

χ

′′ ≤ ≤ ≤ +

for every graph G∈T. For a graph G∈T, let

θ

( )

G =

{

t G∈Tt

}

.

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

x V G

( )

,

S

[ ]

α

,

x

is a

(

dG

( )

x +1

)

-interval,

or

[ ] [ ]

1, \

t

S

α

,

x

is a

(

tdG

( )

x −1

)

-interval. A graph 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 Ft the set of graphs for which there exists a

cyclically interval total t-coloring. Let

F

=

t1

F

t. 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

w G

τc

( )

and

W

τc

( )

G

, respectively. For a

graph G∈F, let Θ

( )

G =

{

t G∈Ft

}

.

It is clear that for any t∈, Tt ⊆Ft and T⊆F. Note that for an arbitrary

graph G,

θ

( )

G

⊆ Θ

( )

G

. It is also clear that for any G∈T, the following

inequality is true

( )

G w Gτc

( )

w Gτ

( )

W Gτ

( )

Wτc

( )

G V G

( )

E G

( )

.

χ

′′ ≤ ≤ ≤ ≤ ≤ +

A middle graph

M G

( )

of a graph G is the graph whose vertex set is

( )

( )

V G

E G

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 Cn , let

V C

( ) {

n

=

v v

1

, ,

2

,

v

n

}

and

( ) {

n 1

, ,

2

,

n

}

E C

=

e e

e

, where e1=v v1 n and ei=v vi−1 i for i=2, 3,,n. For

example, the graphs in Figure 1 are C4 and

M C

( )

4 , respectively. Note that in

Section 3 we always use the kind of diagram like (c) in Figure 1 to denote

( )

n
(3)
[image:3.595.214.539.73.162.2]

DOI: 10.4236/ojdm.2017.74018 202 Open Journal of Discrete Mathematics (a) (b) (c)

Figure 1. C4 and M C( )4 . (a) C4; (b) M C( )4 ; (c) Another diagram of M C( )4 .

2.

C

n

In this section we study the cyclically interval total colorings of

C

n

(

n

3

)

, show that Cn∈F, get the exact values of

( )

c n

w C

τ and

W

τc

( )

C

n , and determine the

set

Θ

( )

G

. In [4] it was proved the following result.

Theorem 1. (H. P. Yap [4]) For any integer n≥3,

( )

3, if 0 mod 3 ;

(

)

4, otherwise.

n

n C

χ

′′ =  ≡

In [5] Petrosyan et al. studied the interval total colorings of cycles and provided the following result.

Theorem 2. (P. A. Petrosyan et al. [5]) For any integer n≥3, we have

1) Cn∈T,

2)

( )

3, if 0 mod 3 ;

(

)

4, otherwise,

n

n w Cτ =  ≡

3)

W C

τ

( )

n

= +

n

2

.

Now we consider the cyclically interval total colorings of

C

n

(

n

3

)

. In order to define the total coloring of the graph Cn easily, we denote

V C

( )

n

E C

( )

n

by

{

a a

1

,

2

,

,

a

2n

}

, where a2 1i− =vi and a2i=ei for any

i

[ ]

1,

n

.

Theorem 3. For any integer n≥3, we have

1) Cn∈F,

2)

( )

3, if 0 mod 3 ;

(

)

4, otherwise,

c n

n w Cτ =  ≡

3)

W

τc

( )

C

n

=

2

n

.

Proof. Since T⊆F, then for any G∈T we have

χ

′′

( )

G

w G

τc

( )

w G

τ

( )

.

So by Theorems 1 and 2, (1) and (2) hold.

Let us prove (3). Now we show that

W

τc

( )

C

n

2

n

for any n≥3. Define a total coloring α of the graph Cn as follows: Let

( )

a

i

i i

,

[ ]

1, 2 .

n

α

=

It is easy to check that α is a cyclically interval total 2n-coloring of Cn. Thus,

( )

2

c n

W

τ

C

n

for any integer n≥3. On the other hand, it is easy to see that

( )

( )

( )

2

c

n n n

Wτ CV C + E C = n. So we have

W

τc

( )

C

n

=

2

n

.

Lemma 4. For any integer n≥3 and

t

[

4, 2

n

2

]

, Cn∈Ft.

Proof. For any

t

[

4,

n

2

]

, we define a total t-coloring α of the graph

n

(4)

DOI: 10.4236/ojdm.2017.74018 203 Open Journal of Discrete Mathematics

Case 1.

n

=

3 ,

k k

.

Subcase 1.1.

t

=

3 ,

s s

[

2, 2

k

1

]

. Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

, if 1, ];

1, if 1, 2 , 1 mod 3 ; 2, if 1, 2 , 2 mod 3 ; 3, if 1, 2 , 0 mod 3 .

i

i i t

i t n i a

i t n i i t n i

α

 ∈  ∈ +  =  ∈ +   ∈ + ≡ 

Subcase 1.2.

t

= +

3

s

1,

s

[

1, 2

k

1

]

. Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

4, if 1; 3, if 2; 2, if 3;

, if 4, ;

1, if 1, 2 , 2 mod 3 ; 2, if 1, 2 , 0 mod 3 ; 3, if 1, 2 , 1 mod 3 .

i

i i i

i i t

a

i t n i i t n i i t n i

α =   =   =  =   ∈ +  ∈ + ≡   ∈ +

Subcase 1.3.

t

= +

3

s

2,

s

[

1, 2

k

2

]

. Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

, if 1; , if 2, ;

1, if 1, 2 , 0 mod 3 ; 2, if 1, 2 , 1 mod 3 ; 3, if 1, 2 , 2 mod 3 .

i

t i

i i t

i t n i a

i t n i i t n i

α =    ∈ + =   ∈ +  ∈ + ≡ 

Case 2.

n

=

3

k

+

1,

k

.

Subcase 2.1.

t

=

3 ,

s s

[

2, 2

k

]

.

Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

4, if 1; 3, if 2; 2, if 3;

, if 4, ;

1, if 1, 2 , 1 mod 3 ; 2, if 1, 2 , 2 mod 3 ; 3, if 1, 2 , 3 mod 3 .

i

i i i

i i t

a

i t n i i t n i i t n i

α =   =   =  =   ∈ +  ∈ + ≡   ∈ +

Subcase 2.2.

t

= +

3

s

1,

s

[

1, 2

k

1

]

. Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

4, if 1; , if 2, ;

1, if 1, 2 , 2 mod 3 ; 2, if 1, 2 , 0 mod 3 ; 3, if 1, 2 , 1 mod 3 .

i

i

i i t

i t n i a

i t n i i t n i

(5)

DOI: 10.4236/ojdm.2017.74018 204 Open Journal of Discrete Mathematics

Subcase 2.3.

t

= +

3

s

2,

s

[

1, 2

k

1

]

. Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

, if 1, ;

1, if 1, 2 , 0 mod 3 ; 2, if 1, 2 , 1 mod 3 ; 3, if 1, 2 , 2 mod 3 .

i

i i t

i t n i a

i t n i i t n i

α

 ∈  ∈ +  =  ∈ +   ∈ + ≡ 

Case 3.

n

=

3

k

+

2,

k

.

Subcase 3.1.

t

=

3 ,

s s

[

2, 2

k

]

.

Let

( )

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

, if 1; , if 2, ;

1, if 1, 2 , 1 mod 3 ; 2, if 1, 2 , 2 mod 3 ; 3, if 1, 2 , 3 mod 3 .

i

t i

i i t

i t n i a

i t n i i t n i

α =    ∈ + =   ∈ +  ∈ + ≡ 

Subcase 3.2.

t

= +

3

s

1,

s

[ ]

1, 2

k

.

Let

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

, if 1, ;

1, if 1, 2 , 2 mod 3 ; ( )

2, if 1, 2 , 0 mod 3 ; 3, if 1, 2 , 1 mod 3 .

i

i i t

i t n i a

i t n i i t n i

α

 ∈  ∈ +  =  ∈ +   ∈ +

Subcase 3.3.

t

= +

3

s

2,

s

[ ]

1, 2

k

.

Let

[ ]

[

]

(

)

[

]

(

)

[

]

(

)

4, if 1; 3, if 2; 2, if 3;

, if 4, ; ( )

1, if 1, 2 , 0 mod 3 ; 2, if 1, 2 , 1 mod 3 ; 3, if 1, 2 , 2 mod 3 .

i

i i i

i i t

a

i t n i i t n i i t n i

α =   =   =  =   ∈ +  ∈ + ≡   ∈ +

It is not difficult to check that, in each case,

α

is always a cyclically interval total t-coloring of Cn. The proof is complete.

Lemma 5. C3∈F5.

Proof. We define a total 5-coloring

α

of the graph C3 as follows: Let

( )

, if

[ ]

1,5 ; 3, if 6.

i

i i

a

i

α

=  ∈

= 

It is easy to see that α is a cyclically interval total coloring of C3.

Lemma 6. For any integer n≥4, Cn∉F2n−1.

Proof. By contradiction. Suppose that, for any integers n≥4, α is a cyclically

interval total

(

2n−1

)

-coloring of Cn. Then there exist different i j, ∈

[

1, 2n

]

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DOI: 10.4236/ojdm.2017.74018 205 Open Journal of Discrete Mathematics

Without loss of generality, we may assume that α

( )

ai

( )

aj =1. Then for each

[

2, 2 1

]

kn− , there is only one vertex or one edge of Cn 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 i=2n, i.e.,

( )

a

2n

1

α

=

. Then we have 3≤ ≤j 2n−3. Note that a2n=v v1 n. Since α is a

cyclically interval total

(

2n−1

)

-coloring of Cn, then we have

( ) ( )

{

α

v1 ,

α

v v1 2

}

=

{ } {

2,3 , 2, 2n−1 or 2

} {

n−2, 2n−1

}

and

( ) (

)

{

α

vn ,

α

vn−1vn

}

=

{ } {

2,3 , 2, 2n−1 or 2

} {

n−2, 2n−1 .

}

Because

( ) ( )

{

α

v1 ,

α

v v1 2

}

{

α

( ) (

vn ,

α

vn−1vn

)

}

= ∅,

without loss of generality, we may assume that

( ) ( )

{

α

v1 ,

α

v v1 2

}

=

{ }

2,3

and

( ) (

)

{

α

vn ,

α

vn−1vn

}

=

{

2n−2, 2n−1 .

}

Since that for each k

[

2, 2n−1

]

there is only one vertex or one edge of Cn is

colored by k. Then α

( )

aj−1 ∈

[

4, 2n−3

]

or α

( )

aj+1 ∈

[

4, 2n−3

]

. On the other

hand, since

α

is a cyclically interval total

(

2n−1

)

-coloring of Cn , then

( ) ( )

aj1 , aj1

{

2,3, 2n 2, 2n 1

}

α − α + ∈ − − weather j is odd or even. A contradiction.

Case 2. i and j are all odd.

Without loss of generality, suppose that i=1.Then we have 3≤ ≤ −j n 1.

Note that ai and aj are all vertices of Cn. Since

α

is a cyclically interval

total

(

2

n

1

)

-coloring of Cn, then we have

( ) ( )

{

α

a2 ,

α

a2n

}

=

{ } {

2,3 , 2, 2n−1 or 2

} {

n−2, 2n−1

}

and

( ) ( )

{

α aj−1 ,α aj+1

}

=

{ } {

2, 3 , 2, 2n−1 or 2

} {

n−2, 2n−1 .

}

Because

( ) ( )

{

α a2 ,α a2n

}

{

α

( ) ( )

aj−1 ,α aj+1

}

= ∅,

without loss of generality, we may assume that

( ) ( )

{

α

a2 ,

α

a2n

}

=

{ }

2,3

and

( ) ( )

{

α aj−1 ,α aj+1

}

=

{

2n−2, 2n−1 ,

}

say α

( )

a2 =2. Then α

( )

a2n =3. Now we consider the color of a3. By the

definition of α , α

( ) {

a3 ∈ 1,3, 4, 2n−1

}

. But

α

( )

a

3 can not be 1, 3 or 2n−1
(7)

DOI: 10.4236/ojdm.2017.74018 206 Open Journal of Discrete Mathematics

that just one vertex or one edge of Cn is colored by

i

, where i

[

2, 2n−1

]

.

Since we already have α

( )

a2n =3 before.

Combining Theorem 3, Corollaries 4 - 6, the following result holds. Theorem 7. For any integer n≥3,

( )

[

[

]

]

{

}

(

)

[

]

{

}

3, 2 , if 3;

3, 2 \ 2 1 , if 4 and 0 mod 3 ; 4, 2 \ 2 1 , otherwise.

n

n n

C n n n n

n n

 =

Θ = − ≥ ≡

3.

M C

( )

n

In this section we study the cyclically interval total colorings of M C

( ) (

n n≥3

)

,

prove M C

( )

n ∈F, get the exact values of

(

( )

)

c n

w M Cτ , provide a lower bound

of c

(

( )

)

n

Wτ M C , and show that for any k between c

(

( )

)

n

w M Cτ and the lower

bound of c

(

( )

)

n

Wτ M C , M C

( )

n ∈Fk.

Theorem 8. For any integer n≥3, c

(

( )

)

5

n

w M Cτ = .

Proof. Suppose that integer n≥3. Now we define a total 5-coloring α of

the graph M C

( )

n as follows:

Case 1. n is even. Let

( )

v

i

3,

i

[ ]

1,

n

,

α

=

( )

1,

[ ]

[ ]

1, ; 4, 1, ,

i

i n

e

i n

α

=  ∈∈ ◊ 

( )

e v

i i

2,

i

[ ]

1,

n

,

α

=

(

1

)

[ ]

[ ]

1, 1, ; 4, 1, ,

i i

i n

v e

i n

α

+ =  ∈

∈ ◊ 

and

(

1

)

[ ]

[ ]

3, 1, ; 5, 1, ,

i i

i n

e e

i n

α

+ =  ∈∈ ◊ 

where en+1=e1. See Figure 2.

By the definition of α we have

[

,

i

] [ ]

1,3 ,

[ ]

1,

,

S

α

v

=

i

n

[

,

i

] [ ]

2, 4 ,

[ ]

1,

,

[image:7.595.194.530.210.724.2]

S

α

v

=

i

∈ ◊

n

(8)

DOI: 10.4236/ojdm.2017.74018 207 Open Journal of Discrete Mathematics

and

[

,

i

] [ ] [ ]

1,5 ,

1,

.

S

α

e

=

i

n

Case 2. n is odd. Let

( )

3,

[

1, 1 ;

]

4, ,

i

i n v

i n

α

=  ∈ −

= 

( )

1,4,

[

[

1,1, 1 ;1 ;

]

]

2, ,

i

i n

e i n

i n

α

 ∈ −

= ∈ ◊ −

=

( )

2,

[

1, 1 ;

]

3, ,

i i

i n

e v

i n

α

=  ∈=

(

1

)

[

[

]

]

{

}

1, 1, 2 1 ; 4, 1, 2 ,

i i

i n n

v e

i n

α

+ =  ∈ − −

∈ ◊ −



 

( )

v e

n1

5,

α

=

(

1

)

[

[

]

]

3, 1, 1 ; 5, 1, 1 ,

i i

i n

e e

i n

α

+

 ∈ −

=  ∈ ◊

 

and

( )

e e

1 n

4.

α

=

See Figure 3.

By the definition of α we have

[

,

i

] [ ]

1,3 ,

[

1,

2

]

{ }

1 ,

S

α

v

=

i

n

n

[

,

i

] [ ]

2, 4 ,

[

1,

2 ,

]

S

α

v

=

i

∈ ◊

n

[

,

n

] [ ]

3,5 ,

S

α

v

=

and

[

,

i

] [ ] [ ]

1,5 ,

1,

.

S

α

e

=

i

n

Combining Cases 1 and 2, we know that, for any integer n≥3, α is a

cyclically interval total 5-coloring of M C

( )

n . Therefore

( )

(

)

5.

c n

[image:8.595.206.496.66.712.2]

w M Cτ

(9)

DOI: 10.4236/ojdm.2017.74018 208 Open Journal of Discrete Mathematics

On the other hand,

( )

(

)

(

( )

)

1 5.

c

n n

w M Cτ ≥ ∆ M C + = So we have

( )

(

)

5.

c n

w M Cτ =

Theorem 9. For any integer n≥3, Wτc

(

M C

( )

n

)

≥4n.

Proof. Now we define a total 4n-coloring α of the graph M C

( )

n as follows:

Let

( )

v

i

4

i

1,

α

= −

( )

e

i

4

i

3,

α

= −

( )

e v

i i

4

i

2,

α

= −

(

v e

i i 1

)

4 ,

i

α

+

=

and

(

e e

i i1

)

4

i

1,

α

+

= −

where

i

[ ]

1,

n

and en+1=e1. See Figure 4. By the definition of

α

we have

[

,

i

] [

4

2, 4 ,

] [ ]

1,

,

S

α

v

=

i

i i

n

[

,

i

] [

4

5, 4

1 ,

] [ ]

2,

,

S

α

e

=

i

i

i

n

and

[

,

1

] [ ] [

1,3

4

1, 4 .

]

S

α

e

=

n

n

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

( )

n . So we

have

( )

(

)

4 .

c n

Wτ M Cn

Theorem 10. For any integer n≥3 and any

k

[

5, 4

n

]

, M C

( )

n ∈Fk.

Proof. Suppose n≥3 and for any

k

[

5, 4

n

]

. We define a total k-coloring

α of M C

( )

n as follows. First we use the colors 1, 2,,k to color the

vertices and edges of M C

( )

n beginning from e1 by the way used in the proof

of Theorem 9. Now we color the other vertices and edges of M C

( )

n with the [image:9.595.260.494.604.707.2]

colors 1, 2,,k.

(10)

DOI: 10.4236/ojdm.2017.74018 209 Open Journal of Discrete Mathematics

Case 1. k≡0 mod 4

(

)

.

Let

4

k

t= . Then we have α

(

v et t+1

)

=k, where 1≤ ≤t n.

Subcase 1.1.

n t

is even.

Let

( )

v

t i

3,

i

[

1,

n t

]

,

α

+

=

( )

1,

[

[

1,

]

]

; 4, 1, ,

t i

i n t

e

i n t

α

+ =  ∈∈ ◊ 

(

e v

t i t i

)

2,

i

[

1,

n t

]

,

α

+ +

=

(

1

)

[

[

]

]

1, 1, ; 4, 1, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈∈ ◊ 

and

(

1

)

[

[

]

]

3, 1, ; 5, 1, ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ −

∈ ◊ −

 

where en+1=e1. See Figure 5.

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [ ]

1, ,

S

α

v

=

i

i i

t

[

,

t i

] [ ]

1,3 ,

[

1,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

2, 4 ,

[

1,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[ ] [

,

i

4

5, 4

1 ,

] [ ]

2, ,

S

α

e

=

i

i

i

t

[

,

t1

] [ ] [

1,3

1,

]

,

S

α

e

+

=

k

k

and

[

,

t i

] [ ] [

1,5 ,

2,

]

.

S

α

e

+

=

i

n t

Subcase 1.2.

n t

is odd.

Let

( )

v

t i

3,

i

[

1,

n t

1 ,

]

α

+

=

− −

( )

v

n

2,

α

=

( )

1,

[

[

1,

]

]

; 4, 1, ,

t i

i n t

e

i n t

α

+ =  ∈ −

∈ ◊ −

[image:10.595.205.519.61.732.2]

 

(11)

DOI: 10.4236/ojdm.2017.74018 210 Open Journal of Discrete Mathematics

(

e v

t i t i

)

2,

i

[

1,

n t

1 ,

]

α

+ +

=

− −

(

e v

n n

)

3,

α

=

(

1

)

[

[

]

]

1, 1, ; 4, 1, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈∈ ◊ 

(

1

)

[

[

]

]

3, 1, 1 ; 5, 1, 1 ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ − −

∈ ◊ − −

 

( )

e e

n 1

2,

α

=

[image:11.595.209.533.71.733.2] [image:11.595.300.444.75.200.2]

and recolor e1 and e v1 1 as α

( )

e1 =4 and α

( )

e v1 1 =5, where en+1=e1. See

Figure 6.

By the definition of

α

we have

[

,

1

] [ ]

3,5 ,

S

α

v

=

[

,

i

] [

4

2, 4 ,

] [ ]

2, ,

S

α

v

=

i

i i

t

[

,

t i

] [ ]

1,3 ,

[

1,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

2, 4 ,

[

1,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[ ] [

,

i

4

5, 4

1 ,

] [ ]

2, ,

S

α

e

=

i

i

i

t

[

,

t1

] [ ] [

1,3

1,

]

,

S

α

e

+

=

k

k

and

[

,

t i

] [ ] [

1,5 ,

2,

]

.

S

α

e

+

=

i

n t

Case 2. k≡1 mod 4

(

)

.

Let 3

4

k

t= + . Then we have

α

( )

e

t

=

k

, where 2≤ ≤t n.

Subcase 2.1.

n t

is even.

Let

( )

v

t

k

,

α

=

( )

v

t i

3,

i

[

1,

n t

]

,

α

+

=

( )

1,

[

[

1,

]

]

;

4, 1, ,

t i

i n t

e

i n t

α

+ =  ∈∈ ◊ 

(12)

DOI: 10.4236/ojdm.2017.74018 211 Open Journal of Discrete Mathematics

( )

e v

t t

1,

α

=

(

e v

t i t i

)

2,

i

[

1,

n t

]

,

α

+ +

=

(

v e

t t 1

)

k

1,

α

+

= −

(

1

)

[

[

]

]

1, 1, ; 4, 1, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈∈ ◊ 

(

e e

t t 1

)

k

,

α

+

=

(

1

)

[

[

]

]

3, 1, ; 5, 1, ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ −

∈ ◊ −

 

and recolor et as α

( )

et =2, where en+1=e1. See Figure 7.

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [

1,

1 ,

]

S

α

v

=

i

i i

t

[

,

t

]

{

1,

1,

}

,

S

α

v

=

k

k

[

,

t i

] [ ]

1,3 ,

[

1,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

2, 4 ,

[

1,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[ ] [

,

i

4

5, 4

1 ,

] [

2,

1 ,

]

S

α

e

=

i

i

i

t

[

,

t

] [ ] [

1, 2

2,

]

,

S

α

e

=

k

k

[

,

t1

] [ ] [

1,3

1,

]

,

S

α

e

+

=

k

k

and

[

,

t i

] [ ] [

1,5 ,

2,

]

.

S

α

e

+

=

i

n t

Subcase 2.2.

n t

is odd.

Let

( )

v

t

1,

α

=

( )

v

t i

3,

i

[

1,

n t

]

,

α

+

=

( )

4,

[

[

1,

]

]

; 1, 1, ,

t i

i n t

e

i n t

α

+ =  ∈ −

∈ ◊ −

 

[image:12.595.203.522.65.719.2]

(

e v

t i t i

)

2,

i

[

0,

n t

]

,

α

+ +

=

(13)

DOI: 10.4236/ojdm.2017.74018 212 Open Journal of Discrete Mathematics

(

v e

t t 1

)

3,

α

+

=

(

1

)

[

[

]

]

4, 1, ; 1, 1, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈ −

∈ ◊ −



(

e e

t t 1

)

1,

α

+

=

(

1

)

[

[

]

]

5, 1, ; 3, 1, ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ −

∈ ◊ −

 

where en+1=e1. See Figure 8.

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [

1,

1 ,

]

S

α

v

=

i

i i

t

[

,

t i

] [ ]

2, 4 ,

[

0,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

1,3 ,

[

0,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[

,

i

] [

4

5, 4

1 ,

] [

2,

1 ,

]

S

α

e

=

i

i

i

t

[ ] [ ] [

,

t

1, 2

2,

]

,

S

α

e

=

k

k

and

[

,

t i

] [ ] [

1,5 ,

1,

]

.

S

α

e

+

=

i

n t

Case 3. k≡2 mod 4

(

)

.

Let 2

4

k

t= + . Then we have

α

( )

e v

t t

=

k

, where 2≤ ≤t n.

Subcase 3.1.

n t

is even.

Let

( )

v

t

k

,

α

=

( )

v

t i

3,

i

[

1,

n t

]

,

α

+

=

( )

1,

[

[

1,

]

]

;

4, 1, ,

t i

i n t

e

i n t

α

+ =  ∈ −

∈ ◊ −

 

(

e v

t i t i

)

2,

i

[

1,

n t

]

,

α

+ +

=

(

v e

t t 1

)

k

1,

α

+

= −

(

1

)

[

[

]

]

1, 1, ; 4, 1, ,

t i t i

i n t

v e

i n t

α

+ + +

 ∈ −

=  ∈ ◊



[image:13.595.204.525.65.719.2]

(14)

DOI: 10.4236/ojdm.2017.74018 213 Open Journal of Discrete Mathematics

(

e e

t t 1

)

k

,

α

+

=

(

1

)

[

[

]

]

3, 1, ; 5, 1, ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ −

∈ ◊ −

 

and recolor e vt t as α

( )

e vt t =1, where en+1=e1. See Figure 9.

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [

1,

1 ,

]

S

α

v

=

i

i i

t

[

,

t

]

{

1,

1,

}

,

S

α

v

=

k

k

[

,

t i

] [ ]

1,3 ,

[

1,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

2, 4 ,

[

1,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[ ] [

,

i

4

5, 4

1 ,

] [

2,

1 ,

]

S

α

e

=

i

i

i

t

[

,

t

]

{ }

1

[

3,

]

,

S

α

e

=

k

k

[

,

t1

] [ ] [

1,3

1,

]

,

S

α

e

+

=

k

k

and

[

,

t i

] [ ] [

1,5 ,

2,

]

.

S

α

e

+

=

i

n t

Subcase 3.2.

n t

is odd.

Let

( )

v

t

1,

α

=

( )

v

t 1

2,

α

+

=

( )

v

t i

3,

i

[

2,

n t

]

,

α

+

=

( )

4,

[

[

1,

]

]

; 1, 1, ,

t i

i n t

e

i n t

α

+ =  ∈∈ ◊ 

(

e v

t1 t 1

)

3,

α

+ +

=

(

e v

t i t i

)

2,

i

[

2,

n t

]

α

+ +

=

(

v e

t t 1

)

2,

α

+

=

(

1

)

[

[

]

]

4, 1, ; 1, 1, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈∈ ◊  [image:14.595.203.523.70.720.2]

(15)

DOI: 10.4236/ojdm.2017.74018 214 Open Journal of Discrete Mathematics

(

e e

t t 1

)

1,

α

+

=

(

1

)

[

[

]

]

5, 1, ; 3, 1, ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ −

∈ ◊ −

 

where en+1=e1. See Figure 10.

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [

1,

1 ,

]

S

α

v

=

i

i i

t

[

,

t

]

{

1, 2,

}

,

S

α

v

=

k

[

,

t i

] [ ]

2, 4 ,

[

1,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

1,3 ,

[

1,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[

,

i

] [

4

5, 4

1 ,

] [

2,

1 ,

]

S

α

e

=

i

i

i

t

[

,

t

]

{ }

1

[

3,

]

,

S

α

e

=

k

k

and

[

,

t i

] [ ] [

1,5 ,

1,

]

.

S

α

e

+

=

i

n t

Case 4. k≡3 mod 4

(

)

.

Let 1

4

k

t= + . Then we have

α

( )

v

t

=

k

, where 2≤ ≤t n.

Subcase 4.1.

n t

is even.

Let

( )

v

t i

3,

i

[

1,

n t

]

,

α

+

=

( )

4,

[

[

1,

]

]

;

1, 1, ,

t i

i n t

e

i n t

α

+ =  ∈ −

∈ ◊ −

 

(

e v

t i t i

)

2,

i

[

1,

n t

]

,

α

+ +

=

(

1

)

[

[

]

]

4, 0, ; 1, 0, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈∈ ◊ 

(

1

)

[

[

]

]

3, 1, ; 5, 1, ,

t i t i

i n t

e e

i n t

α

+ + +

 ∈ −

=  ∈ ◊

 

[image:15.595.198.534.60.741.2]

and recolor e1 as

α

( )

e

1

=

4

, where en+1=e1. See Figure 11.
(16)
[image:16.595.206.536.58.737.2]

DOI: 10.4236/ojdm.2017.74018 215 Open Journal of Discrete Mathematics

Figure 11. A total 7-coloring of M C

( )

6 .

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [

1,

1 ,

]

S

α

v

=

i

i i

t

[

,

t

]

{

1,

1,

}

,

S

α

v

=

k

k

[

,

t i

] [ ]

2, 4 ,

[

1,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

1,3 ,

[

1,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[ ] [

,

i

4

5, 4

1 ,

] [ ]

2, ,

S

α

e

=

i

i

i

t

[

,

t1

] [ ]

1, 4

{ }

,

S

α

e

+

=

k

[

,

t i

] [ ] [

1,5 ,

2,

]

.

S

α

e

+

=

i

n t

Subcase 4.2.

n t

is odd.

Let

( )

v

t 1

4,

α

+

=

( )

v

t i

3,

i

[

2,

n t

]

,

α

+

=

( )

e

t 1

2,

α

+

=

( )

4,

[

[

2,

]

]

;

1, 2, ,

t i

i n t

e

i n t

α

+ =  ∈ −

∈ ◊ −

 

(

e v

t1 t 1

)

3,

α

+ +

=

(

e v

t i t i

)

2,

i

[

2,

n t

]

,

α

+ +

=

(

v e

t t 1

)

1,

α

+

=

(

v e

t 1 t 2

)

5,

α

+ +

=

(

1

)

[

[

]

]

4, 2, ; 1, 2, ,

t i t i

i n t

v e

i n t

α

+ + + =  ∈∈ ◊ 

(

e e

t 1 t 2

)

4,

α

+ +

=

(

1

)

[

[

]

]

5, 2, ; 3, 2, ,

t i t i

i n t

e e

i n t

α

+ + + =  ∈ −

∈ ◊ −

 

(17)
[image:17.595.210.533.61.447.2]

DOI: 10.4236/ojdm.2017.74018 216 Open Journal of Discrete Mathematics

Figure 12. A total 7-coloring of M C

( )

5 .

By the definition of α we have

[

,

i

] [

4

2, 4 ,

] [

1,

1 ,

]

S

α

v

=

i

i i

t

[

,

t

]

{

1,

1,

}

,

S

α

v

=

k

k

[

,

t 1

] [ ]

3,5 ,

S

α

v

+

=

[

,

t i

] [ ]

2, 4 ,

[

2,

]

,

S

α

v

+

=

i

n t

[

,

t i

] [ ]

1,3 ,

[

2,

]

,

S

α

v

+

=

i

∈ ◊

n t

[

,

1

] [ ]

1,5 ,

S

α

e

=

[ ] [

,

i

4

5, 4

1 ,

] [ ]

2, ,

S

α

e

=

i

i

i

t

[

,

t1

] [ ]

1, 4

{ }

,

S

α

e

+

=

k

and

[

,

t i

] [ ] [

1,5 ,

2,

]

.

S

α

e

+

=

i

n t

Combining Cases 1 - 4, the result follows.

4. Concluding Remarks

In this paper, we study the cyclically interval total colorings of cycles and middle graphs of cycles.

For any integer n≥3, we show Cn∈F, prove that

w C

τc

( )

n

=

3

(if

(

)

0 mod 3

n

) or 4 (otherwise) and

W

τc

( )

C

n

=

2

n

, and determine the set

Θ

( )

G

as

( )

[

[

]

]

{

}

(

)

[

]

{

}

3, 2 , if 3;

3, 2 \ 2 1 , if 4 and 0 mod 3 ; 4, 2 \ 2 1 , otherwise.

n

n n

C n n n n

n n

 =

Θ = − ≥ ≡

For any integer n≥3, we have

M C

( )

n

F

, prove that w M Cτc

(

( )

n

)

=5

and Wτc

(

M C

( )

n

)

≥4n and, for any k between 5 and 4n,

M C

( )

n

F

k. We

conjecture that Wτc

(

M C

( )

n

)

=4n.

It would be interesting in future to study the cyclically interval total colorings of graphs related to cycles.

Acknowledgements

(18)

DOI: 10.4236/ojdm.2017.74018 217 Open Journal of Discrete Mathematics

supported in part by the Natural Science Foundation of Hebei Province of China under Grant A2015106045, and in part by the Institute of Applied Mathematics of Shijiazhuang University.

References

[1] Vizing, V.G. (1965) Chromatic Index of Multigraphs. Doctoral Thesis, Novosibirsk. (in Russian)

[2] Behzad, M. (1965) Graphs and Their Chromatic Numbers. Ph.D. Thesis, Michigan State University, East Lansing, MI.

[3] Petrosyan, P.A. (2007) Interval Total Colorings of Complete Bipartite Graphs. Pro-ceedings of the CSIT Conference,Yerevan, 84-85.

[4] Yap, H.P. (1996) Total Colorings of Graphs, Lecture Notes in Mathematics 1623. Springer-Verlag , Berlin.

http://www.scirp.org/journal/ojdm : 10.4236/ojdm.2017.74018 http://creativecommons.org/licenses/by/4.0/

Figure

Figure 1. C  and 4M C()4. (a) C ; (b) 4M C()4; (c) Another diagram of M C()4.
Figure 2. A total 5-coloring of M C()4.
Figure 3. A total 5-coloring of M C()5.
Figure 4. A total 16-coloring of M C()4.
+7

References

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