Cyclically Interval Total Coloring of the One Point Union of Cycles

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ISSN Online: 2161-7643 ISSN Print: 2161-7635

DOI: 10.4236/ojdm.2018.83006 May 18, 2018 65 Open Journal of Discrete Mathematics

Cyclically Interval Total Coloring of the One

Point Union of Cycles

Shijun Su

1

, Wenwei Zhao

2

, Yongqiang Zhao

3*

1_{School of Science, Hebei University of Technology, Tianjin, China}

2_{School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei, China} 3_{School of Science, Shijiazhuang University, Shijiazhuang, China}

Abstract

A total coloring of a graph G with colors 1,2, , t is called a cyclically inter-val total t-coloring if all colors are used, and the edges incident to each vertex

( )

v V G∈ _{together with}_v_{are colored by}

(

d vG

( )

+1

)

consecutive colors

modulo t, where d vG

( )

is the degree of the vertex v in G. The one point un-ion ( )k

n

C of k-copies of cycle Cn is the graph obtained by taking v as a common vertex such that any two distinct cycles Cn′ and Cn′′ are edge dis-joint 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 n≥3_and k≥2_.

Keywords

Total Coloring, Interval Total Coloring, Cyclically Interval Total Coloring, Cycle, One Point Union of Cycles

1. Introduction

We denote the sets of vertices and edges in a graph G by V G

( )

and E G

( )

, respectively. For a vertex x V G∈

( )

_{, we denote the degree of}_x_in_G_by d x_G

( )

, and we use ∆

( )

G _{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 ele-ment 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 E G V G

( )



( )

to 

How to cite this paper: Su, S.J., Zhao, W.W. and Zhao, Y.Q. (2018) Cyclically Interval Total Coloring of the One Point Union of Cycles. Open Journal of Discrete Mathematics, 8, 65-72.

https://doi.org/10.4236/ojdm.2018.83006

Received: March 26, 2018 Accepted: May 15, 2018 Published: May 18, 2018

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DOI: 10.4236/ojdm.2018.83006 66 Open Journal of Discrete Mathematics

α of a graph G and for any v V G∈

( )

_{, let}

[ ]

,

{

( )

}

{

( )

| is incident to

}

S

α

v =

α

v 

α

e e v .

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}

(

d xG

( )

+1

)

-interval. A graph G is

interval total colorable if it has an interval total t-coloring for some positive in-teger 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 x V G∈

( )

_, S

[ ]

α

,x _is

a

(

d xG

( )

+1

)

-interval, or

[ ] [ ]

1, \t S

α

,x is a

(

t d x− G

( )

−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=



t≥₁Ft. For any graph G∈F, the minimum and the maximum values of t for which G has a cyclically interval to-tal t-coloring are denoted by _{w G}c

( )

τ and W Gτc

( )

, respectively. It is clear that for any G∈F, the following inequality is true:

( )

_G _{w G W G}c

( )

c

( )

_{V G}

( )

_{E G}

( )

.

τ τ

χ

′′ ≤ ≤ ≤ +

The one point union ( )k n

C of k-copies of cycle Cn is the graph obtained by taking v as a common vertex such that any two distinct cycles Cn′ and Cn′′ are edge disjoint and do not have any vertex in common except v. In this paper, we study the cyclically interval total colorability of ( )k

n

C . Let

( )

( )k k₁

( )

i

n i n

V C =

_

₌V C

and V C

( ) {

ni = v v1i, , ,2i  vni

}

, where Cni is the i-th copy of Cn and i∈

[ ]

1,k . Without loss of generality, we may assume that the common vertex v of the

k-copies of cycle Cn is the first vertex in each cycle, i.e., v v v= =11 12= = v1k. For example, the graphs in Figure 1 are all ( )3

6

C . Note that in the paper we al-ways use the kind of diagram like (b) in Figure 1 to denote ( )k

n

C .

All graphs considered in this paper are finite undirected simple graphs.

2. Main Results

Vaidya and Isaac [5] studied the total coloring of ( )k n

C and got the following result.

Theorem 1 (Vaidya and Isaac) For any integers n≥3 _and k≥2 _,

( )

k 2 1

n

C k

χ

′′ = + .

Now we consider the cyclically interval total colorings of ( )k n

C , show that ( )k

n

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DOI: 10.4236/ojdm.2018.83006 67 Open Journal of Discrete Mathematics (a) (b)

Figure 1. (a) ( )3

6

C ; (b) Another diagram of ( )3 6

C .

( )

k c

n

W Cτ .

Theorem 2 For any integers n≥3_and k≥2_, c

( )

( )k 2 1

n

w Cτ = k+ .

Proof. Suppose that n≥3 and k≥2_{. Let}

( )

( )k k₁ i

n i n

V C =



₌C , where i n

C is the i-th copy of Cn. Let V C

( ) {

ni = v v1i, , ,2i  vni

}

, where i∈

[ ]

1,k . 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., 1 2

1 1 1k

v v v= = = = v . Now we define a total

(

2k+1

)

_-coloring_α_{of the graph} ( )k

n

C as follows: Case 1. n≡0 mod3

(

)

_.

Let

( )

v 1,

α

=

( )

[ ]

(

)

[ ]

(

)

2 1, 2, , 1, and 1 mod3 ; 2 , 2, , 1, and 0 mod3 ; 2 1, 2, , 1, and 2 mod3 ,

j i

j i n j k i

v j i n j k i

j i n j k i

α

 − ∈ ∈ ≡



= ∈ ∈ ≡

 + ∈ ∈ ≡



(

)

[ ]

(

)

[ ]

(

)

1

2 1, 1, , 1, and 2 mod3 ; 2 , 1, , 1, and 1 mod3 ; 2 1, 1, , 1, and 0 mod3 ,

j j i i

j i n j k i

v v j i n j k i

j i n j k i

α +

 − ∈ ∈ ≡



=_ ∈ ∈ ≡

 + ∈ ∈ ≡



where vnj+1=v1j =v for any j∈

[ ]

1,k . See Figure 2 for an example. By the definition of α, we have

[ ] [

, 1,2 1 ;

]

S

α

v = k+

[

]

[ ]

, j 2 1,2 1 , 2, and 1, .

i

S_

α

v  =_ j− j+ i∈ n j∈ k

This shows that α is a cyclically interval total

(

2k+1

)

_{-coloring of} ( )k n

C .

Case 2. n≡1 mod3

(

)

_.

Let

( )

v 1,

α

=

( )

[

]

[ ]

(

)

[

]

[ ]

(

)

[

]

[ ]

(

)

{

}

[ ]

2 1, 2, 3 , 1, and 1 mod3 ; 2 , 2, 3 , 1, and 0 mod3 ; 2 1, 2, 3 , 1, and 2 mod3 ;

2 2, 2, and 1, ;

2 1, 1 and 1, ,

j i

j i n j k i

v j i n j k i

j i n n j k

j i n j k

α

 − ∈ − ∈ ≡

 _∈ ₋ _∈ _≡



=_ + ∈ − ∈ ≡

 + ∈ − ∈



− = − ∈

[image:3.595.218.532.65.200.2]

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[image:4.595.203.538.53.761.2]

Figure 2. 7-total coloring of ( )3

6 C .

(

)

[

]

[ ]

(

)

[

]

[ ]

(

)

[

]

[ ]

(

)

{

}

[ ]

1

2 1, 1, 3 , 1, and 2 mod3 ; 2 , 1, 3 , 1, and 1 mod3 ; 2 1, 1, 3 , 1, and 0 mod3 ,

2 1, 2, and 1, ;

2 , = 1 and 1, ,

j j i i

j i n j k i

v v j i n j k i

j i n n j k

j i n j k

α +  − ∈ − ∈ ≡  _∈ ₋ _∈ _≡  =_ + ∈ − ∈ ≡  + ∈ − ∈  − ∈ 

[ ]

1,k . Recolor vnk−2,v vnk−1, nk and v vnk−2 nk−1 as

( )

vnk 2 2k 2,

( )

vnk1 2k 1,

( )

vnk 2 1k

α

− = −

α

− = +

α

= − and

α

(

v vnk−2 nk−1

)

=2 1k− . See

Figure 3 for an example.

By the definition of α, we have

[ ] [

, 1,2 1 ;

]

S

α

v = k+

[

]

[

]

{

}

[ ]

, j 2 1,2 1 , 2, 3 1 and 1, ;

i

S_

α

v _ = j− j+ i∈ n−  n− j∈ k

[

]

{

}

[

]

, j 2 ,2 2 , 2, and 1, 1 ;

i

S_

α

v  =_ j j+ i∈ −n n j∈ k−

[

]

2

, k 2 2,2 ;

n

S

α

v−  = k− k

[

]

, k 2 1,2 1 .

n

S_

α

v  =_ k− k+

(

2k+1

)

C .

Case 3. n≡2 mod3

(

)

_.

Let

( )

v 1,

α

=

( )

[

]

[ ]

(

)

[

]

[ ]

(

)

[

]

[ ]

(

)

[ ]

2 1, 2, 1 , 1, and 1 mod3 ; 2 , 2, 1 , 1, and 0 mod3 ; 2 1, 2, 1 , 1, and 2 mod3 ; 2 2, and 1, ,

j i

j i n j k i

v

j i n j k i

j i n j k

α

 − ∈ − ∈ ≡  _∈ ₋ _∈ _≡  =  ₊ _∈ ₋ _∈ _≡   + = ∈ 

(

)

[

] [

]

[ ]

(

)

[

] [

]

[ ]

(

)

[

] [

]

[ ]

(

)

[ ]

1

2 1, 1, 4 2, 1 , 1, and 2 mod3 ; 2 , 1, 4 2, 1 , 1, and 1 mod3 ; 2 1, 1, 4 2, 1 , 1, and 0 mod3 ; 2 2, 3 and 1, ;

2 1, and 1, ,

j j i i

j i n n n j k i

v v j i n n n j k i

j i n j k

α ₊  − ∈ − − − ∈ ≡  _∈ ₋ ₋ ₋ _∈ _≡  =_ + ∈ − − − ∈ ≡  ₊ _{= −} _∈  + = ∈    

[ ]

1,k . Recolor vnk−2,v v v vnk−1, ,nk nk−3 nk−2,v vnk−2 nk−1 and 1

k k

n n

v v− as

α

( )

vnk−2 =2k−2,

α

( )

vnk−1 =2k+1,

α

( )

vnk =2k,

(

v vnk3 nk2

)

2 1k

[image:4.595.261.495.68.165.2]

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7

[image:5.595.222.510.64.546.2]

C .

8

C .

an example.

[ ] [

, 1,2 1 ;

]

S

α

v = k+

[

]

[

]

{

}

[ ]

, j 2 1,2 1 , 2, 4 1 and 1, ;

i

S_

α

v _ = j− j+ i∈ n−  n− j∈ k

[

] {

}

[

]

, j 2 ,2 2 , 3, 2, and 1, 1 ;

i

S_

α

v  =_ j j+ i∈ −n n− n j∈ k−

[

]

3

, k 2 1,2 1 ;

n

S

α

v−  = k− k+

[

]

2

, k 2 2,2 ;

n

S

α

v−  = k− k

[

]

, k 2 1,2 1 .

n

S_

α

v  =_ k− k+

(

2k+1

)

C .

Combining Cases 1-3, we have c

( )

( )k 2 1

n

w Cτ ≤ k+ . On the other hand, by

Theorem 1, c

( )

( )k

( )

( )k 2 1

n n

w Cτ ≥

χ

′′ C = k+ . So we have w Cτc

( )

n( )k =2k+1. Theorem 3 For any integers n≥3_and k≥2_,

( )

₍

2₂ ₂

₎

1, ₂ _1, 2₂ _1.2; 2 2

k c

n

n k k n

W C _n k _{n k} _k _n

n τ

+ − ≤ −

 

≥  ₋  ₊ _{+ −} _≥ ₋

 _ ₋ _



Proof. Suppose that n≥3_and k≥2_{. We consider the following two cases.}

Case 1. k≤2n−2_.

Now we define a total

(

2n k+ −1

)

_-coloring_α_{of the graph} ( )k n

C as follows: Let

( )

v 1,

α

=

( )

j 2 2,

[ ]

2, and

[ ]

1, ;

i

v i j i n j k

α

= + − ∈ ∈

(

v vi ij j1

)

2i j 1,i

[ ]

1, andn j

[ ]

1, ,k

α

+ = + − ∈ ∈

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[image:6.595.204.537.48.801.2]

4

C .

[ ] [

, 1, 1

] [

2 ,2 1 ;

]

S

α

v = k+ _ n n k+ −

[

]

[ ]

, j 2 3,2 1 , 2, and 1, .

i

S_

α

v  =_ i j+ − i j+ − i∈ n j∈ k

(

2n k+ −1

)

C .

So we have c

( )

( )k 2 1

n

W Cτ ≥ n k+ − if k≤2n−2.

Case 2. k≥2n−1_.

Let 2 2 j j s n  

= _ ₋ _ and tj= −j

(

2n−2

)

sj. Now we define a total

(

2n k+ −1

)

_-coloring_α_{of the graph} ( )k n

C as follows: Let

( )

v 1,

α

=

( )

(

)

(

)

[ ]

4 4 2 2

2 2 2 2, 2, and 1, ;

2 2

j

i j j

v n s i t

j

n i j i n j k

n

α = − + + −

  = − _ _+ + − ∈ ∈ −  

(

)

(

)

(

)

[ ]

1 4 4 2 1

2 2 2 1, 1, and 1, ,

2 2

j j

i i j j

v v n s i t

j

n i j i n j k

n

α + = − + + −

 

= − _ _+ + − ∈ ∈

−

 

[ ]

1,k . See Figure 6 for an example. By the definition of α, we have

[ ]

(

)

(

)

(

)

(

)

(

)

(

)

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

1, 2 2 1

2 2

4 4 2 , 2 2 2 1 ;

2 2 2 2

k k k k

S v n s t n s n n s n t

k

n k

n

k k

n n n n k

n n

α =_ − + +  _{ } − + − + + − _

    =_ − _ _+ + _ −      ₋  ₊ ₋  ₊ _{+ −}   _ ₋ _ _ ₋ _     

(

)

(

)

(

)

(

)

[ ]

, 4 4 2 3, 4 4 2 1

2 2 2 3, 2 2 2 1 ,

2 2 2 2

2, and 1, .

j

i j j j j

S v n s i t n s i t

j j

n i j n i j

n n

i n j k

α   =_ − + + − − + + − _         =_ − _ _+ + − − _ _+ + − _ − −       ∈ ∈

(

2 2

)

2 1 2 2

k

n n k

n

 ₋  ₊ _{+ −} 

 _ ₋ _ 

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[image:7.595.263.487.74.294.2]

4

C .

-coloring of ( )k n

C . So we have

( )

(

2 2

)

2 1 2 2

k c

n k

W C n n k

n

τ ≥ −  ₋ + + −

  for any

2 1

k≥ n− _.

3. Generalization

The one point of union _C( )k _{of any}_k_cycles

1 2

1_, 2_{, ,}

k k

n n n

C C  C is the graph ob-tained by taking v as a common vertex such that any two distinct cycles i

i n

C

and j_j n

C are edge disjoint and do not have any vertex in common except v. By the proof of Theorem 2, the following definitions are well defined.

Definition 4 A partial

(

i i, 1+

)

_{-total coloring of}C_n

(

n≥3

)

_{is a coloring}

( )

( ) { }

1

[

]

:V Cn E Cn \ v i 1, 2i

α

_ → − + _{such that}

α

( )

v v_{1 2} =i _,

α

( )

v vn ₁ = +i 1

and S

α

,vj is an interval for each j∈

[ ]

2,n . A partial

(

i i, 1+

)

′-total

color-ing of Cn

(

n≥3

)

is a coloring

α

′:V C

( )

n E C

( ) {

n \ ,v v1 n

}

→ −

[

i 2, 1i+

]

such that

α

′

( )

v v1 2 =i,

α

′

( )

v vn 1 = +i 1 and S

α

′,vj is an interval for each

[ ]

2,

j∈ n _.

Now we consider the cyclically interval total colorings of _C( )k _.

Theorem 5 For any integer k≥2_, _{w C}c

( )

( )k 2_k 1

τ = + .

Proof. Suppose that graph _C( )k _{is the one point of union of cycles}

1 2

1_, 2_{, ,}

k k

n n n

C C  C . Let

( ) {

i 1, , ,2 i

}

i i i i

n n

V C = v v _ v _{, where}i∈

[ ]

1,k _{. Without loss of}

generality, we may assume that the common vertex v of the k cycles i_i n

C is the first vertex in each cycle, i.e., 1 2

1 1 1k

v v v= = = =_ v _{. Now we define a total}

(

2 1k+

)

_-coloring_α_{of the graph}_C( )k _{as follows: Let}

_α

( )

_v ₌₁_,

i ni C

α

be a partial

(

2 ,2 1i i+

)

_{-total coloring of} i_i

n

C for each i∈

[

1,k−1

]

_{, and} k nk C

α

be a

partial

(

2 ,2 1k k+

)

′-total coloring of k_k n

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DOI: 10.4236/ojdm.2018.83006 72 Open Journal of Discrete Mathematics In this section, we consider the one point of union _C( )k _of_k_{cycles with}

dif-ferent length, show that C( )k ∈F, get the exact values of w Cτc

( )

( )k , and the further research maybe more interesting.

Acknowledgements

We thank the editor and the referee for their valuable comments. The work was 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.

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

[4] Zhao, Y. and Su, S. (2017) Cyclically Interval Total Colorings of Cycles and Middle Graphs of Cycles. Open Journal of Discrete Mathematics, 7, 200-217.

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