Oct 31, 2017 - A total coloring of a graph G is a function ... A k -interval is a set of k consecutive integers. .... T . For a graph GâT , the least and the greatest.
Open Journal of Discrete Mathematics, 2017, 7, 200-217 http://www.scirp.org/journal/ojdm ISSN Online: 2161-7643 ISSN Print: 2161-7635
Cyclically Interval Total Colorings of Cycles and Middle Graphs of Cycles Yongqiang Zhao1, Shijun Su2 School of Science, Shijiazhuang University, Shijiazhuang, China School of Science, Hebei University of Technology, Tianjin, China
1 2
How to cite this paper: Zhao, Y.Q. and Su, S.J. (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 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). http://creativecommons.org/licenses/by/4.0/ Open Access
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 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 i, i = 1, 2, , t , and for
{
}
any x ∈ V ( G ) , the set S [α , v ] = {α ( v )} α ( e ) e is incident to v
( dG ( x ) + 1) -interval,
or
{1, 2, , t} \ S [α , x ]
is a
is a
( t − dG ( x ) − 1) -interval,
where dG ( 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 dG ( 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 . DOI: 10.4236/ojdm.2017.74018 Oct. 31, 2017
200
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
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 ) e is incident to 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
( 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≥1Tt . For a graph G ∈ T , the least and the greatest
values of t for which G ∈ Tt are denoted by wτ ( G ) and Wτ ( G ) , respectively. Clearly,
χ ′′ ( G ) ≤ wτ ( G ) ≤ Wτ ( G ) ≤ V ( G ) + E ( G ) G) for every graph G ∈ T . For a graph G ∈ T , let θ (=
{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
( t − dG ( 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 = t≥1Ft . 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 wτ ( G ) and Wτ ( 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 c
c
inequality is true
χ ′′ ( G ) ≤ wτc ( G ) ≤ wτ ( G ) ≤ Wτ ( G ) ≤ Wτc ( G ) ≤ V ( G ) + E ( G ) 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 ( Cn ) = {v1 , v2 , , vn } and E ( Cn ) = {e1 , e2 , , en } , where e1 = v1vn and ei = vi−1vi for i = 2,3, , n . For example, the graphs in Figure 1 are C4 and M ( C4 ) , respectively. Note that in Section 3 we always use the kind of diagram like (c) in Figure 1 to denote
M ( Cn ) .
DOI: 10.4236/ojdm.2017.74018
201
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
(a)
(b)
(c)
Figure 1. C4 and M ( C4 ) . (a) C4 ; (b) M ( C4 ) ; (c) Another diagram of M ( C4 ) .
2. Cn In this section we study the cyclically interval total colorings of Cn ( n ≥ 3) , show
that Cn ∈ F , get the exact values of wτ ( Cn ) and Wτ ( Cn ) , and determine the c
c
set Θ ( G ) . In [4] it was proved the following result. Theorem 1. (H. P. Yap) For any integer n ≥ 3 ,
3, if n ≡ 0 ( mod 3) ; χ ′′ ( Cn ) = 4, otherwise In [5] Petrosyan et al. studied the interval total colorings of cycles and provided the following result. Theorem 2. (P. A. Petrosyan et al. T-K) For any integer n ≥ 3 , we have 1) Cn ∈ T ,
3, if n ≡ 0 ( mod 3) ; 2) wτ ( Cn ) = 4, otherwise 3) Wτ ( Cn )= n + 2 .
Now we consider the cyclically interval total colorings of Cn ( n ≥ 3) . In order
to define the total coloring of the graph Cn easily, we denote V ( Cn ) E ( Cn ) by
a2i−1 = vi and a2i = ei for any i ∈ [1, n ] . Theorem 3. For any integer n ≥ 3 , we have 1) Cn ∈ F ,
{a1 , a2 , , a2n } , where
3, if n ≡ 0 ( mod 3) ; c 2) wτ ( Cn ) = 4, otherwise
3) Wτ ( Cn ) = 2n . c Proof. Since T ⊆ F , then for any G ∈ T we have χ ′′ ( G ) ≤ wτ ( G ) ≤ wτ ( G ) . So by Theorems 1 and 2, (1) and (2) hold. c Let us prove (3). Now we show that Wτ ( Cn ) ≥ 2n for any n ≥ 3 . Define a total coloring α of the graph Cn as follows: Let c
α ( a= i ) i, i ∈ [1, 2n ] It is easy to check that α is a cyclically interval total 2n-coloring of Cn . c Thus, Wτ ( Cn ) ≥ 2n for any integer n ≥ 3 . On the other hand, it is easy to see c 2n . So we have Wτc ( Cn ) = 2n . that Wτ ( Cn ) ≤ V ( Cn ) + E ( Cn ) = Lemma 4. For any integer n ≥ 3 and t ∈ [ 4, 2n − 2] , Cn ∈ Ft . Proof. For any t ∈ [ 4, n − 2] , we define a total t-coloring α of the graph Cn as follows: DOI: 10.4236/ojdm.2017.74018
202
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
Case 1.= n 3k , k ∈ . Subcase 1.1. t =3s, s ∈ [ 2, 2k − 1] . Let
i, 1, α ( ai ) = 2, 3,
if i ∈ [1, t ]]; if i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) ; if i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) ; if i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) .
Subcase 1.2. t = 3s + 1, s ∈ [1, 2k − 1] . Let
4, 3, 2, α ( ai ) = i, 1, 2, 3,
if i = 1; if i = 2; if i = 3;
if i ∈ [ 4, t ] ;
if i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) ; if i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) ; if i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) .
Subcase 1.3. t =3s + 2, s ∈ [1, 2k − 2] . Let
t , i, α ( ai ) = 1, 2, 3,
if if if if if
i = 1; i ∈ [ 2, t ] ; i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) .
Case 2. n = 3k + 1, k ∈ . t 3s, s ∈ [ 2, 2k ] . Subcase 2.1.= Let
4, 3, 2, α ( ai ) = i, 1, 2, 3,
if if if if if if if
i = 1; i = 2; i = 3; i ∈ [ 4, t ] ; i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 3 ( mod 3) .
Subcase 2.2. t = 3s + 1, s ∈ [1, 2k − 1] . Let
4, i, α ( ai ) = 1, 2, 3, DOI: 10.4236/ojdm.2017.74018
203
if if if if if
i = 1; i ∈ [ 2, t ] ; i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) . Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
Subcase 2.3. t =3s + 2, s ∈ [1, 2k − 1] . Let
i, 1, α ( ai ) = 2, 3,
if i ∈ [1, t ] ; if i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) ; if i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) ; if i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) .
Case 3. n =3k + 2, k ∈ . t 3s, s ∈ [ 2, 2k ] . Subcase 3.1.= Let
t , i, α ( ai ) = 1, 2, 3,
if if if if if
i = 1; i ∈ [ 2, t ] ; i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 3 ( mod 3) .
Subcase 3.2. t = 3s + 1, s ∈ [1, 2k ] . Let
i, 1, α (ai ) = 2, 3,
if if if if
i ∈ [1, t ] ; i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) .
Subcase 3.3. t =3s + 2, s ∈ [1, 2k ] . Let
4, 3, 2, α (ai ) = i, 1, 2, 3,
if if if if if if if
i = 1; i = 1; i = 1; i ∈ [ 4, t ] ; i ∈ [t + 1, 2n ] , i ≡ 0 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 1 ( mod 3) ; i ∈ [t + 1, 2n ] , i ≡ 2 ( mod 3) .
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 i, if i ∈ [1,5] ; α ( ai ) = 3, if i = 6. It is easy to see that α is a cyclically interval total coloring of C3 . Lemma 6. For any integer n ≥ 4 , Cn ∉ F2 n−1 .
Proof. By contradiction. Suppose that, for any integers n ≥ 4 , α is a cyclically ( 2n − 1) -coloring of Cn . Then there exist different i, j ∈ [1, 2n] such that α ( ai ) = α ( a j ) and for different s, t ∈ [1, 2n ] \ {i, j} , α ( ai ) ≠ α ( a j ) .
interval total
DOI: 10.4236/ojdm.2017.74018
204
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
( )
a j 1 . Then for each Without loss of generality, we may assume that α= ( ai ) α=
k ∈ [ 2, 2n − 1] , 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., α ( a2 n ) = 1 . Then we have 3 ≤ j ≤ 2n − 3 . Note that a2 n = v1vn . Since α is a cyclically interval total ( 2n − 1) -coloring of Cn , then we have
{α ( v1 ) ,α ( v= 1v2 )} {2,3} , {2, 2n − 1} or {2n − 2, 2n − 1} and
v )} {2,3} , {2, 2n − 1} or {2n − 2, 2n − 1} {α ( v ) ,α ( v = n−1 n
n
Because
{α ( v1 ) , α ( v1v2 )} {α ( vn ) ,α ( vn−1vn )} = ∅ without loss of generality, we may assume that
{α ( v1 ) ,α ( v1v2 )} = {2,3} and
{α ( v ) ,α ( v
v
n−1 n
n
)} ={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 a j −1 ∈ [ 4, 2n − 3] or α a j +1 ∈ [ 4, 2n − 3] . On the other
( )
( )
hand, since α is a cyclically interval total ( 2n − 1) -coloring of Cn , then α a j −1 , α a j +1 ∈ {2,3, 2n − 2, 2n − 1} 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 a j are all vertices of Cn . Since α is a cyclically interval total ( 2n − 1) -coloring of Cn , then we have
a )} {2,3} , {2, 2n − 1} or {2n − 2, 2n − 1} {α ( a ) ,α (= 2
2n
and
a )} {α ( a ) ,α ( = j −1
j +1
{2,3} , {2, 2n − 1} or {2n − 2, 2n − 1}
Because
{α ( a ) , α ( a )} {α ( a ) , α ( a )} = ∅ j −1
2n
2
j +1
without loss of generality, we may assume that
{α ( a ) ,α ( a )} = {2,3} 2n
2
and
{α ( a ) ,α ( a )} ={2n − 2, 2n − 1} j −1
j +1
say α ( a2 ) = 2 . Then α ( a2 n ) = 3 . Now we consider the color of a3 . By the definition of α , α ( a3 ) ∈ {1,3, 4, 2n − 1} . But α ( a3 ) can not be 1, 3 or 2n − 1
obviously. So we have α ( a3 ) = 4 , and then α ( a4 ) = 3 . This is a contradiction
DOI: 10.4236/ojdm.2017.74018
205
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
to that just one vertex or one edge of Cn is colored by i, where i ∈ [ 2, 2n − 1] . Since we already have α ( a2 n ) = 3 before. Combining Theorem 3, Corollaries 4 - 6, the following result holds. Theorem 7. For any integer n ≥ 3 ,
[3, 2n ] , if n = 3; Θ = ( Cn ) [3, 2n] \ {2n − 1} , if n ≥ 4 and n ≡ 0 ( mod 3) ; [ 4, 2n ] \ {2n − 1} , otherwise.
3. M ( C n ) In this section we study the cyclically interval total colorings of M ( Cn ) ( n ≥ 3) , c prove M ( Cn ) ∈ F , get the exact values of wτ ( M ( Cn ) ) , provide a lower bound c c of Wτ ( M ( Cn ) ) , and show that for any k between wτ ( M ( Cn ) ) and the lower c bound of Wτ ( M ( Cn ) ) , M ( Cn ) ∈ Fk . c Theorem 8. For any integer n ≥ 3 , wτ ( M ( Cn ) ) = 5 . Proof. Suppose that integer n ≥ 3 . Now we define a total 5-coloring α of the graph M ( Cn ) as follows: Case 1. n is even. Let
α ( v= 3, i ∈ [1, n ] i) 1, i ∈ [1, n ] ; α ( ei ) = 4, i ∈ ◊ [1, n ] ,
2, i ∈ [1, n ] α ( ei v= i) 1, i ∈ [1, n ] ; α ( vi ei+1 ) = 4, i ∈ ◊ [1, n ] , and
3, i ∈ [1, n ] ; α ( ei ei+1 ) = 5, i ∈ ◊ [1, n ] , where en+1 = e1 . See Figure 2. By the definition of α we have
S [= α , vi ] [1,3] , i ∈ [1, n ]
α , vi ] S [=
[ 2, 4] , i ∈ ◊ [1, n]
Figure 2. A total 5-coloring of DOI: 10.4236/ojdm.2017.74018
206
M ( C4 ) . Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
and
S [= α , ei ] [1,5] , i ∈ [1, n ] Case 2. n is odd. Let
3, i ∈ [1, n − 1] ; α ( vi ) = 4, i = n, 1, i ∈ [1, n − 1] ; α= ( ei ) 4, i ∈ ◊ [1, n − 1] ; 2, i = n,
2, i ∈ [1, n − 1] ; α ( ei vi ) = 3, i = n, 1, i ∈ [1, n − 2] {n − 1} ; α ( vi ei+1 ) = 4, i ∈ ◊ [1, n − 2] ,
α ( vn e1 ) = 5 3, i ∈ [1, n − 1] ; α ( ei ei+1 ) = 5, i ∈ ◊ [1, n − 1] , and
α ( e1en ) = 4 See Figure 3. By the definition of α we have
S [α , vi ] = [1,3] , i ∈ [1, n − 2] {n − 1} S [= α , vi ]
[ 2, 4] , i ∈ ◊ [1, n − 2] S [α , vn ] = [3,5]
and
S [= α , ei ] [1,5] , i ∈ [1, n ] Combining Cases 1 and 2, we know that, for any integer n ≥ 3 , α is a cyclically interval total 5-coloring of M ( Cn ) . Therefore
wτc ( M ( Cn ) ) ≤ 5
Figure 3. A total 5-coloring of M ( C5 ) . DOI: 10.4236/ojdm.2017.74018
207
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
On the other hand,
wτc ( M ( Cn ) ) ≥ ∆ ( M ( Cn ) ) + 1 =5
So we have
wτc ( M ( Cn ) ) = 5 Theorem 9. For any integer n ≥ 3 , Wτ ( M ( Cn ) ) ≥ 4n . Proof. Now we define a total 4n-coloring α of the graph M ( Cn ) as follows: Let c
α ( vi =) 4i − 1 α ( ei =) 4i − 3
α ( ei vi =) 4i − 2
α ( vi ei+1 ) = 4i and
α ( ei ei+1=) 4i − 1 where i ∈ [1, n ] and en+1 = e1 . See Figure 4. By the definition of α we have
S [α , vi ] =− [ 4i 2, 4i ] , i ∈ [1, n]
S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, n ] and
= S [α , e1 ] [1,3] [ 4n − 1, 4n ] This shows that α is a cyclically interval total 4n-coloring of M ( Cn ) . So we have
Wτc ( M ( Cn ) ) ≥ 4n Theorem 10. For any integer n ≥ 3 and any k ∈ [5, 4n ] , M ( Cn ) ∈ Fk . Proof. Suppose n ≥ 3 and for any k ∈ [5, 4n ] . We define a total k-coloring α of M ( Cn ) as follows. First we use the colors 1, 2, , k to color the vertices and edges of M ( Cn ) beginning from e1 by the way used in the proof of Theorem 9. Now we color the other vertices and edges of M ( Cn ) with the colors 1, 2, , k .
Figure 4. A total 16-coloring of M ( C4 ) . DOI: 10.4236/ojdm.2017.74018
208
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
Case 1. k ≡ 0 ( mod 4 ) . k Let t = . Then we have α ( vt et +1 ) = k , where 1 ≤ t ≤ n . 4 Subcase 1.1. n − t is even. Let
α ( vt +i ) =3, i ∈ [1, n − t ]
1, i ∈ [1, n − t ] ; α ( et +i ) = 4, i ∈ ◊ [1, n − t ] ,
α ( et +i vt +i ) =2, i ∈ [1, n − t ] 1, i ∈ [1, n − t ] ; α ( vt +i et +i+1 ) = 4, i ∈ ◊ [1, n − t ] , and
3, i ∈ [1, n − t ] ; α ( et +i et +i+1 ) = 5, i ∈ ◊ [1, n − t ] ,
where en+1 = e1 . See Figure 5. By the definition of α we have
S [α , vi ] =− [ 4i 2, 4i ] , i ∈ [1, t ]
S [α , vt +i ] = [1,3] , i ∈ [1, n − t ]
, vt +i ] [ 2, 4] , i ∈ ◊ [1, n − t ] S [α= S [α , e1 ] = [1,5]
S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t ] S= [α , et +1 ] [1,3] [ k − 1, k ] and
S [α , et +i ]= [1,5] , i ∈ [ 2, n − t ]
Subcase 1.2. n − t is odd. Let
α ( vt +i )= 3, i ∈ [1, n − t − 1]
α ( vn ) = 2 1, i ∈ [1, n − t ] ; α ( et +i ) = 4, i ∈ ◊ [1, n − t ] ,
Figure 5. A total 8-coloring of M ( C6 ) . DOI: 10.4236/ojdm.2017.74018
209
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
α ( et +i vt +i )= 2, i ∈ [1, n − t − 1]
α ( en vn ) = 3 1, i ∈ [1, n − t ] ; α ( vt +i et +i+1 ) = 4, i ∈ ◊ [1, n − t ] , 3, i ∈ [1, n − t − 1] ; α ( et +i et +i+1 ) = 5, i ∈ ◊ [1, n − t − 1] ,
α ( en e1 ) = 2 and recolor e1 and e1v1 as α ( e1 ) = 4 and α ( e1v1 ) = 5 , where en+1 = e1 . See Figure 6. By the definition of α we have
S [α , v1 ] = [3,5]
S [α , vi ] =− [ 4i 2, 4i ] , i ∈ [ 2, t ] S [α , vt +i ] = [1,3] , i ∈ [1, n − t ] S [α= , vt +i ] [ 2, 4] , i ∈ ◊ [1, n − t ]
S [α , e1 ] = [1,5] S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t ] S= [α , et +1 ] [1,3] [ k − 1, k ] and
S [α , et +i ]= [1,5] , i ∈ [ 2, n − t ] Case 2. k ≡ 1( mod 4 ) . k +3 Let t = . Then we have α ( et ) = k , where 2 ≤ t ≤ n . 4 Subcase 2.1. n − t is even. Let
α ( vt ) = k α ( vt +i ) =3, i ∈ [1, n − t ] 1, i ∈ [1, n − t ] ; α ( et +i ) = 4, i ∈ ◊ [1, n − t ] ,
Figure 6. A total 8-coloring of M ( C5 ) . DOI: 10.4236/ojdm.2017.74018
210
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
α ( et vt ) = 1 α ( et +i vt +i ) =2, i ∈ [1, n − t ] α ( vt et +1 )= k − 1 1, i ∈ [1, n − t ] ; α ( vt +i et +i+1 ) = 4, i ∈ ◊ [1, n − t ] ,
α ( et et +1 ) = k 3, i ∈ [1, n − t ] ; α ( et +i et +i+1 ) = 5, i ∈ ◊ [1, n − t ] , and recolor et as α ( et ) = 2 , where en+1 = e1 . See Figure 7. By the definition of α we have
S [α , vi ] =[ 4i − 2, 4i ] , i ∈ [1, t − 1]
S [α ,= vt ] {1, k − 1, k} S [α , vt +i ] = [1,3] , i ∈ [1, n − t ] , vt +i ] [ 2, 4] , i ∈ ◊ [1, n − t ] S [α= S [α , e1 ] = [1,5]
S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t − 1] = S [α , et ] [1, 2] [ k − 2, k ]
S= [α , et +1 ] [1,3] [ k − 1, k ] and
S [α , et +i ]= [1,5] , i ∈ [ 2, n − t ]
Subcase 2.2. n − t is odd. Let
α ( vt ) = 1 α ( vt +i ) =3, i ∈ [1, n − t ] 4, i ∈ [1, n − t ] ; α ( et +i ) = 1, i ∈ ◊ [1, n − t ] ,
α ( et +i vt +i ) =2, i ∈ [ 0, n − t ]
Figure 7. A total 9-coloring of M ( C5 ) . DOI: 10.4236/ojdm.2017.74018
211
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
α ( vt et +1 ) = 3 4, i ∈ [1, n − t ] ; α ( vt +i et +i+1 ) = 1, i ∈ ◊ [1, n − t ] , α ( et et +1 ) = 1 5, i ∈ [1, n − t ] ; α ( et +i et +i+1 ) = 3, i ∈ ◊ [1, n − t ] , where en+1 = e1 . See Figure 8. By the definition of α we have
S [α , vi ] =[ 4i − 2, 4i ] , i ∈ [1, t − 1] S [α , vt +i ]= [ 2, 4] , i ∈ [ 0, n − t ]
S [α= , vt +i ] [1,3] , i ∈ ◊ [ 0, n − t ] S [α , e1 ] = [1,5] S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t − 1]
= S [α , et ] [1, 2] [ k − 2, k ] and
S [α , et +i =] [1,5] , i ∈ [1, n − t ]
Case 3. k ≡ 2 ( mod 4 ) . k +2 Let t = . Then we have α ( et vt ) = k , where 2 ≤ t ≤ n . 4 Subcase 3.1. n − t is even. Let
α ( vt ) = k α ( vt +i ) =3, i ∈ [1, n − t ] 1, i ∈ [1, n − t ] ; α ( et +i ) = 4, i ∈ ◊ [1, n − t ] , α ( et +i vt +i ) =2, i ∈ [1, n − t ]
α ( vt et +1 )= k − 1 1, i ∈ [1, n − t ] ; α ( vt +i et +i+1 ) = 4, i ∈ ◊ [1, n − t ] ,
Figure 8. A total 9-coloring of M ( C6 ) . DOI: 10.4236/ojdm.2017.74018
212
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
α ( et et +1 ) = k 3, i ∈ [1, n − t ] ; α ( et +i et +i+1 ) = 5, i ∈ ◊ [1, n − t ] , and recolor et vt as α ( et vt ) = 1 , where en+1 = e1 . See Figure 9. By the definition of α we have
S [α , vi ] =[ 4i − 2, 4i ] , i ∈ [1, t − 1]
S [α ,= vt ] {1, k − 1, k} S [α , vt +i ] = [1,3] , i ∈ [1, n − t ] S [α= , vt +i ] [ 2, 4] , i ∈ ◊ [1, n − t ] S [α , e1 ] = [1,5]
S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t − 1] S= [α , et ] {1} [ k − 3, k ]
S= [α , et +1 ] [1,3] [ k − 1, k ] and
S [α , et +i ]= [1,5] , i ∈ [ 2, n − t ]
Subcase 3.2. n − t is odd. Let
α ( vt ) = 1 α ( vt +1 ) = 2
α ( vt +i ) =3, i ∈ [ 2, n − t ] 4, i ∈ [1, n − t ] ; α ( et +i ) = 1, i ∈ ◊ [1, n − t ] ,
α ( et +1vt +1 ) = 3
α ( et +i vt +i ) =2, i ∈ [ 2, n − t ] α ( vt et +1 ) = 2 4, i ∈ [1, n − t ] ; α ( vt +i et +i+1 ) = 1, i ∈ ◊ [1, n − t ] ,
Figure 9. A total 10-coloring of M ( C5 ) . DOI: 10.4236/ojdm.2017.74018
213
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
α ( et et +1 ) = 1 5, i ∈ [1, n − t ] ; α ( et +i et +i+1 ) = 3, i ∈ ◊ [1, n − t ] , where en+1 = e1 . See Figure 10. By the definition of α we have
S [α , vi ] =[ 4i − 2, 4i ] , i ∈ [1, t − 1] S [α , vt ] = {1, 2, k} S [α , vt +i ]=
[ 2, 4] , i ∈ [1, n − t ] S [α= , vt +i ] [1,3] , i ∈ ◊ [1, n − t ] S [α , e1 ] = [1,5] S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t − 1] S= [α , et ] {1} [ k − 3, k ] and
S [α , et +i =] [1,5] , i ∈ [1, n − t ] Case 4. k ≡ 3 ( mod 4 ) . k +1 Let t = . Then we have α ( vt ) = k , where 2 ≤ t ≤ n . 4 Subcase 4.1. n − t is even. Let
α ( vt +i ) =3, i ∈ [1, n − t ] 4, i ∈ [1, n − t ] ; α ( et +i ) = 1, i ∈ ◊ [1, n − t ] ,
α ( et +i vt +i ) =2, i ∈ [1, n − t ] 4, i ∈ [ 0, n − t ] ; α ( vt +i et +i+1 ) = 1, i ∈ ◊ [ 0, n − t ] , 3, i ∈ [1, n − t ] ; α ( et +i et +i+1 ) = 5, i ∈ ◊ [1, n − t ] , and recolor e1 as α ( e1 ) = 4 , where en+1 = e1 . See Figure 11.
Figure 10. A total 10-coloring of M ( C6 ) . DOI: 10.4236/ojdm.2017.74018
214
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
Figure 11. A total 7-coloring of M ( C6 ) .
By the definition of α we have
S [α , vi ] =[ 4i − 2, 4i ] , i ∈ [1, t − 1] S [α ,= vt ] {1, k − 1, k} S [α , vt +i ]=
[ 2, 4] , i ∈ [1, n − t ] S [α= , vt +i ] [1,3] , i ∈ ◊ [1, n − t ] S [α , e1 ] = [1,5] S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t ] S [α , et +1 ] = [1, 4] {k} S [α , et +i ]= [1,5] , i ∈ [ 2, n − t ] Subcase 4.2. n − t is odd. Let
α ( vt +1 ) = 4
α ( vt +i ) =3, i ∈ [ 2, n − t ] α ( et +1 ) = 2 4, i ∈ [ 2, n − t ] ; α ( et +i ) = 1, i ∈ ◊ [ 2, n − t ] ,
α ( et +1vt +1 ) = 3
α ( et +i vt +i ) =2, i ∈ [ 2, n − t ] α ( vt et +1 ) = 1
α ( vt +1et +2 ) = 5 4, i ∈ [ 2, n − t ] ; α ( vt +i et +i+1 ) = 1, i ∈ ◊ [ 2, n − t ] ,
α ( et +1et +2 ) = 4 5, i ∈ [ 2, n − t ] ; α ( et +i et +i+1 ) = 3, i ∈ ◊ [ 2, n − t ] , where en+1 = e1 . See Figure 12. DOI: 10.4236/ojdm.2017.74018
215
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
Figure 12. A total 7-coloring of M ( C5 ) .
By the definition of α we have
S [α , vi ] =[ 4i − 2, 4i ] , i ∈ [1, t − 1] S [α ,= vt ] {1, k − 1, k} S [α , vt +1 ] = [3,5]
S [α , vt +i ] = [ 2, 4] , i ∈ [ 2, n − t ] S [α= , vt +i ] [1,3] , i ∈ ◊ [ 2, n − t ]
S [α , e1 ] = [1,5] S [α , ei ] =[ 4i − 5, 4i − 1] , i ∈ [ 2, t ] S [α , et +1 ] = [1, 4] {k} and
S [α , et +i ]= [1,5] , i ∈ [ 2, 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 wτ ( Cn ) = 3 (if n ≡ 0 ( mod 3) ) or 4 (otherwise) and Wτc ( Cn ) = 2n , and determine the set Θ ( G ) c
as
[3, 2n ] , if n = 3; Θ = ( Cn ) [3, 2n] \ {2n − 1} , if n ≥ 4 and n ≡ 0 ( mod 3) ; [ 4, 2n ] \ {2n − 1} , otherwise. c For any integer n ≥ 3 , we have M ( Cn ) ∈ F , prove that wτ ( M ( Cn ) ) = 5
c and Wτ ( M ( Cn ) ) ≥ 4n and, for any k between 5 and 4n, M ( Cn ) ∈ Fk . We
c conjecture that Wτ ( M ( Cn ) ) = 4n .
It would be interesting in future to study the cyclically interval total colorings of graphs related to cycles.
Acknowledgements We thank the editor and the referee for their valuable comments. The work was DOI: 10.4236/ojdm.2017.74018
216
Open Journal of Discrete Mathematics
Y. Q. Zhao, S. J. Su
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
DOI: 10.4236/ojdm.2017.74018
[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. Proceedings of the CSIT Conference, Yerevan, 84-85.
[4]
Yap, H.P. (1996) Total Colorings of Graphs, Lecture Notes in Mathematics 1623. Springer-Verlag , Berlin.
[5]
Petrosyan, P.A., Torosyan, A.Yu. and Khachatryan, N.A. (2010) Interval Total Colorings of Graphs . arXiv:1010.2989v1.
217
Open Journal of Discrete Mathematics
