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
21School 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 noadjacent 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(
t−dG( )
x −1)
-interval,where
d
G( )
x
is the degree of the vertex x in G. In this paper, we study thecyclically 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
( )
andE G
( )
denote the set of vertices and edges of G,respectively. For a vertex
x V G
∈
( )
, letd
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 ah
-interval ifD
=
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).
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 noadjacent 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 ofcolors 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 setS
[ ]
α
,
x
is a(
dG( )
x +1)
-interval. A graph G isinterval 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≥1T
t. For a graph G∈T, the least and the greatestvalues of t for which G∈Tt are denoted by
w G
τ( )
andW 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(
t−dG( )
x −1)
-interval. A graph G is cyclically intervaltotal 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≥1F
t. For any graph G∈F, theminimum and the maximum values of t for which G has a cyclically interval total t-coloring are denoted by
w G
τc( )
andW
τc( )
G
, respectively. For agraph 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 followinginequality 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 areadjacent 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. Forexample, the graphs in Figure 1 are C4 and
M C
( )
4 , respectively. Note that inSection 3 we always use the kind of diagram like (c) in Figure 1 to denote
( )
nDOI: 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
nIn 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 andW
τc( )
C
n , and determine theset
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 denoteV C
( )
n
E C
( )
nby
{
a a
1,
2,
,
a
2n}
, where a2 1i− =vi and a2i=ei for anyi
∈
[ ]
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
ii 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
τC
≥
n
for any integer n≥3. On the other hand, it is easy to see that( )
( )
( )
2c
n n n
Wτ C ≤V 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 graphn
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
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]
DOI: 10.4236/ojdm.2017.74018 205 Open Journal of Discrete Mathematics
Without loss of generality, we may assume that α
( )
ai =α( )
aj =1. Then for each[
2, 2 1]
k∈ n− , 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
2n1
α
=
. Then we have 3≤ ≤j 2n−3. Note that a2n=v v1 n. Since α is acyclically 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}
={ }
2,3and
( ) (
)
{
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 iscolored by k. Then α
( )
aj−1 ∈[
4, 2n−3]
or α( )
aj+1 ∈[
4, 2n−3]
. On the otherhand, 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 intervaltotal
(
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}
={ }
2,3and
( ) ( )
{
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 thedefinition of α , α
( ) {
a3 ∈ 1,3, 4, 2n−1}
. Butα
( )
a
3 can not be 1, 3 or 2n−1DOI: 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
( )
nIn 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
(
( )
)
nWτ M C , and show that for any k between c
(
( )
)
nw M Cτ and the lower
bound of c
(
( )
)
nWτ M C , M C
( )
n ∈Fk.Theorem 8. For any integer n≥3, c
(
( )
)
5n
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
i3,
i
[ ]
1,
n
,
α
=
∈
( )
1,[ ]
[ ]
1, ; 4, 1, ,i
i n
e
i n
α
= ∈∈ ◊
( )
e v
i i2,
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
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
n15,
α
=
(
1)
[
[
]
]
3, 1, 1 ; 5, 1, 1 ,
i i
i n
e e
i n
α
+ ∈ −
= ∈ ◊ −
and
( )
e e
1 n4.
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 ≤
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
i4
i
1,
α
= −
( )
e
i4
i
3,
α
= −
( )
e v
i i4
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 ofα
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 wehave
( )
(
)
4 .c n
Wτ M C ≥ n
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-coloringα of M C
( )
n as follows. First we use the colors 1, 2,,k to color thevertices and edges of M C
( )
n beginning from e1 by the way used in the proofof 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.
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 i3,
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
=
[ ] [
,
i4
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 i3,
i
[
1,
n t
1 ,
]
α
+=
∈
− −
( )
v
n2,
α
=
( )
1,[
[
1,]
]
; 4, 1, ,t i
i n t
e
i n t
α
+ = ∈ −∈ ◊ −
[image:10.595.205.519.61.732.2]
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 12,
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. SeeFigure 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
=
[ ] [
,
i4
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
tk
,
α
=
( )
v
t i3,
i
[
1,
n t
]
,
α
+=
∈
−
( )
1,[
[
1,]
]
;4, 1, ,
t i
i n t
e
i n t
α
+ = ∈∈ ◊ −−
DOI: 10.4236/ojdm.2017.74018 211 Open Journal of Discrete Mathematics
( )
e v
t t1,
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
=
[ ] [
,
i4
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
t1,
α
=
( )
v
t i3,
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
+ +=
∈
−
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
−
[ ] [ ] [
,
t1, 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
tk
,
α
=
( )
v
t i3,
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]
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
=
[ ] [
,
i4
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
t1,
α
=
( )
v
t 12,
α
+=
( )
v
t i3,
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]
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 i3,
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.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
=
[ ] [
,
i4
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 14,
α
+=
( )
v
t i3,
i
[
2,
n t
]
,
α
+=
∈
−
( )
e
t 12,
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
+ + + = ∈ −∈ ◊ −
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
=
[ ] [
,
i4
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) andW
τc( )
C
n=
2
n
, and determine the setΘ
( )
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)
=5and Wτc
(
M C( )
n)
≥4n and, for any k between 5 and 4n,M C
( )
n∈
F
k. Weconjecture 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
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.