Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, 268 - 272, 2016
doi:10.21311/001.39.1.28
Edge Chromatic Number on Some Equal Degree Graph
Xinchun Wang
College of Science,North China University of Science and Technology,Tangshan063009,Hebei,China
Wei Li
Institute of Information Engineering,North China University of Science and
Technology,Tangshan063009,Hebei,China
Xiaoying Li
Institute of Electronic and Control Engineering,North China Institute of
AerospaceEngineering,Langfang065000,Hebei,China
Xu Wang
College of Science,North China University of Science and Technology,Tangshan063009,Hebei,China
Abstract
Let G(V , E ) be a simple graph and k is a positive integer, if it exists a mapping of f , and satisfied with
f (e1 )  f (e2 ) ， for two incident edges e1 , e2  E (G), f (e1 )  f (e2 ) ,then f is called the k -proper-edge
coloring of G( K  PEC for short).The minimal number of colors required for a proper edge coloring of G is
denoted by  (G) and is called the proper edge chromatic number. It exists a k -proper edge coloring of simple
graph of G , for any two adjacent vertices u and v in G , the set of colors assigned to the edges incident to u
differs from the set of colorsincident to v ,then f is called k -adjacent-vertex-distinguishing proper edge
coloring, is avvreviated k - AVDEC, also called a adjacentstrong edge coloring. The minimal number of colors
 (G) and is called adjacentrequired for a adjacent-vertex-distinguishing edge coloring of G is denoted by  ad
vertex-distinguishing edge chromatic number.The new class graphs of equal degree graph are be introduced, and
this class graphs adjacent-vertex-distinguishing edge chromatic numbers of path,cycle, fan, complete graph,
wheel, star are presented in this paper.
Key words:Graph Wheel, Join-graph, Adjacent Vertex-distinguishing, Edge Chromatic Number.
1. INTRODUCTION
The coloring problem of graphs is an extremely difficult problem, widely applied in practice. Some
conditional coloring problems are introduced by Harary (Harary,1985). Some network problems can be
converted to the edge coloring (Burris and Schelp,1997;Bazgan C Harkat-Benhamdine and Li
1999;Balister,2003) and adjacent strong edge coloring.
2. PRELIMINARIES
With Definition 1(Bondy J A and Marty U S R,1976) For a graph G(V , E ) , if a proper coloring f is
satisfied with C (u)  C (v), for u, v V (G)(u  v) and u, v  E (G) , then f is called k -adjacent-vertexdistinguishing edge coloring of G , is abbreviated k -AVDEC, and
 (G)  min{k / k  AVDEC of G}
ad
is called the adjacent-vertex-distinguishing edge chromatic number of G [6], where
C (u)  { f (uv) uv  E (G)}.
Conjecture (J.cerny,1996 ) Let G be a connected graph with G  3, and G  C5 (5  cycle), then
ad (G)  (G)  2
where (G) is the maximum degree of graph G .
Definition 2 (Balister and Schelp,2002)Let G(V , E ) be a simple graph, M (G) is called the equal degree
graph of G , where
V (M (G))  E(G)  V   {W }
E(M (G))  E(G)  {uv u V (G), v V , d (u)  d (v)} {wv v V }
Whereas V   {v v V (G)},{w}  (V (G)  V )   .
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, 268 - 272, 2016
3. MAIN RESULT AND CONCLUSIONS
Lemma1 For n  2 , then
Lemma2 For n  3 , then
3, n  2,3
( M ( Pn ))  
 n, n  4
(M (Cn ))  n  2
Lemma3For n  3 , then
 5, n  2,3
( M ( Fn ))  
n  1, n  4
Lemma4 For n  2 , then
 3, n  2

( M ( K n ))   5, n  3
2n  1, n  4

Lemma5 For n  3 , then
 7, n  3
( M (Wn ))  
n  3, n  4
Lemma6For n  3 , then
 3, n  1, 2
( M ( Sn ))  
n  1, n  3
Lemma7 Let G be a connected simple graph with V(G)  3 ,if uv  E (G) and d (u)  d (v)  (G), then
 (G)  (G)  1
ad
Theorem 1 For n  2 , then
5，n＝2，3
 ( M ( pn )＝
xad
 n+1，n  4
Proof Let V ( Pn )  {u1 , u2 ,, un }, V   {v1 , v2 ,, vn } ，there are two cases to be discussed as follow:
Case1 When n = 2, Δ(M(P2)) = 3 from lemma 1. However， u1 , u 2 , v1 , v 2 are vertices which are maximum
 (M ( P2 ))  (M ( Pn ))  4 from lemma 7. If M ( P2 )
degree of M ( P2 ) and all adjacent, ad
exists 4-AVDEC, because four vertices of maximum degree are adjacent each other, and
 4
   4 . The sets are {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}. Let every set correspond a vertex, we can obtain
 3
the vertices of coloring same color are odd, but the vertices of coloring same color are even by graph theory, so
M ( P2 ) don’t exists 4-AVDEC. In order to prove the conclusion be true, we only need construct a map f from
E (M ( P2 )) to {1, 2, 3, 4, 5}:
f (u1v2 )  f (u2 v1 )  1 ; f (u1uv1 )  2 ; f (u2 v2 )  3 ; f (u1u2 )  4; ; f (wv1 )  3 ; f (wv2 )  5 ;
then we have
C (u1 )  {3,5} ; C (u2 )  {2,5} ; C (v1 )  {4,5} ; C (v2 )  {2, 4}
Obviously, the f is 5-AVDEC of M ( P2 ) , the conclusion is true.
When n  3 ,Δ(M(P3)) = 3 from lemma 1. But M ( P2 )  M ( P3 ) , M ( P3 )  5 from n=2. It is obviously to
prove exists 5-AVDEC of M ( P3 ) , we only need to construct a map f from E (M ( P3 )) to {1, 2, 3, 4, 5}:
f (u1u2 )  f (u3v3 )  1 ; f (u1v1 )  f (u2u3 )  2 ; f (u1v3 )  f (wv2 )  3;
f (u2uv2 )  f (wv1 )  4 ; f (u3v1 )  f (wv3 )  5
then we have
C (u1 )  {4,5} ; C (u2 )  {3,5} ; C (u3 )  {3, 4} ; C (v1 )  {1,3} ; C (v2 )  {1, 2,5} ; C (v3 )  {2, 4}
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, 268 - 272, 2016
Obviously, the f is 5  AVDEC of M ( P3 ) , the conclusion is true.
Case2When n ≥ 4, Δ(M(Pn)) = n from lemma 1. However the vertices of u2 , u3 ,, un 1 are vertices which
 (M ( Pn ))  n  1 from lemma 7. It is obviously to prove
are maximum degree, and exist some are adjacent, ad
exists (n  1)  AVDEC of M ( Pn ) , we only need to construct a map f from
E (M ( Pn )) to {1, 2, • • • , n, 0}:
f (ui ui 1 )  i  1, i  1, 2,, n  1 ; f (u1v1 )  1 ; f (u1vn )  2 ;
f (ui u j )  i  j  2(mod n), i  2,3,, n  1, j  2,3,, n  1 ;
f (un v1 )  n ;
f (un vn )  0 ;
f (wvi )  i  1, i  1, 2,, n
so we have
C (u1 )  {0,1, 2} ; C (u2 )  {n} ; C (ui )  {i  3}, i  3, 4,, n  1 ; C (un )  {0, n  2, n  1} ;
C (v1 )  {0,1, n} ; C (vi )  {i  1, i,, n  i  3}(mod n  1) , i  2,3,, n  1 ; C (vn )  {0, 2, n  1} .
Obviously, the f f is (n  1) -AVDEC of M ( Pn ) , the conclusion is true.
From all of above two cases, the conclusion is true.
Theorem 2 For n ≥ 3, then
ad (M (Cn ))  n  3
Proof Let V (Cn )  {u1 , u2 ,, un } , V   {v1 , v2 ,, vn } , u1 , u2 ,, un are vertices of maximum degree and exists
some adjacent, (M (Cn ))  n  3 from lemma 2 and 7. It is obviously to prove exists (n  3) -AVDEC of M (Cn ) ,
we only need to construct a map f from E (M (Cn )) to {1, 2, • • • , n + 2, 0}:
f (ui ui 1 )  i  1, i  1, 2,, n  1 ; f (u1vn )  0 ;
f (ui v j )  i  j (mod n  3), i  1, 2,, n  1, j  1, 2,, n
f (un vi )  n  j, j  1, 2; f (un v j )  j  2, j ,34,, n , f (wv j )  j  1 .
so we have
C (u1 )  {n  2} ; C (ui )  {i  1} , i  2,3,, n  1 ; C (un )  {n}
Obviously, the f is (n  3) -AVDEC of M (Cn ) , the conclusion is true.
Theorem 3 For n ≥ 3, then
 6, n  2,3
n  2, n  4
Proof Let V ( Fn )  {u0 , u1 , u2 ,, un } , V   {v0 , v1 , v2 ,, vn } ,there are three cases to be discussed as follow:
Case1 When n  2, M ( F2 )  M (C3 ), so we can obtain  (M ( F2 ))   (M (C3 ))  6 , then the conclusion is true.
Case2 When n = 3, Δ(M(F3)) = 5 from lemma 3, and M ( F3 ) exists two vertices of maximum degree are
 (M ( F3 ))  6 by lemma 7. It is obviously to prove exists 6-AVDEC of M ( F3 ) , we only need to
adjacent, so ad
construct a map f from E (M ( F3 )) to {1, 2, • • • , 5, 0}:
f (u0ui )  i  1, i  1, 2,3; f (ui ui 1 )  i  1, i  1, 2;
f (u0 v0 )  3；f (u0v2 )  4；f (u1v1 )  3；
f (u1v3 )  4；f (u2v0 )  4；f (u2v2 )  5；
f (u3v1 )  4；f (u3v3 )  5；f (wvi )  i 1, i  0,1, 2,3；
then we have
C (u0 )  {5}, C (u2 )  {0}
Obviously, the f is 6-AVDEC of M ( F3 ) , the conclusion is true.
Case3 When n ≥ 4, Δ(M(Fn)) = n + 1 from lemma 3, M ( Fn ) exists some vertices of maximum degree are
adjacent，  (M ( Fn ))  (M ( Fn ))  1  n  2 by lemma 3,7. It is obviously to prove exists (n  2) -AVDEC of
M ( Fn ) , we only need to construct a map f from E (M ( Fn )) to {1, 2, • • • , n + 1, 0}:
f (u0 vi )  i  1, i  1, 2,n ；f (ui vi 1 )  i  1；i  1, 2,, n;
f (u0 v0 )  n；f (u1v1 )  3；f (u1vn )  4；
f (ui v j )  i  j (mod n  2)；i  2,3,, n  1;
 ' ( M ( Fn ))  
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, 268 - 272, 2016
f (un v1 )  1；f (un vn )  2；f (wvi )  i  1, i  0,1, 2,n；
so we have
C (u0 )  {n  2}, C(ui )  {i  2}, i  2,3,, n 1, C( w)  {0}
Obviously, the f is n  2 -AVDEC of M ( Fn ) , the conclusion is true.
From all of above, the conclusion is true.
Theorem 4 For n ≥ 2, then
5, n  2;

   ( M ( K n ))  6, n  3;
( M ( K ))  1, n  4
n

Proof Let V ( Kn )  u1 , u2 ,, un  ,V   v1 , v2 ,, vn  ，there are three cases to be discussed as follow:
Case1 When n  2 , M ( K2 )  M ( P2 ) , so we can obtain  (M ( K2 ))   (M ( P2 ))  5 , the conclusion is true.
Case2 When n  3 , M ( K3 )  M (C3 ) , so we can obtain  (M ( K3 ))   (M (C3 ))  6 ,the conclusion is true.
Case3 When n  4 , (M ( K n ))  2n 1 from lemma 4, M ( Kn ) exists some vertices of maximum degree are
adjacent,  (M ( Kn ))  (M ( Kn ))  1  2n by lemma 4,7. It is obviously to prove exists 2n -AVDEC of M ( Kn ) ,
we only need to construct a map f from E (M ( K n )) to {1, 2,  , 2n − 1, 0}:
f (ui u j )  i  j  3, i  1, 2,, n  1; j  i  1, i  2,, n;
f (ui v j )  i  j  n  3(mod 2n), i  1, 2,, n; j  1, 2,, n 1;
f (vnu1 )  2n 1, f (vnui )  2(i 1) 1, i  2,3,, n;
f (wvi )  i, i  1, 2,, n 1;
f (wvn )  n, n  0(mod 2) ； f (wvn )  n  1, n  1(mod 2) .
then we have
C (ui )  {2n  i  3}(mod 2n), i  1, 2,, n
Obviously, the f is 2n -AVDEC of M ( Kn ) , the conclusion is true.
From all of above, the conclusion is true.
Theorem 5 For n ≥ 3, then
8, n  3;
n  4, n  4.
 ( M (Wn ))  
Proof Let V (Wn )  {u0 , u1 , u2 ,, un },V   {v0 , v1 , v2 ,, vn } , there are two cases to be discussed as follow:
Case1 When n  3 , M (W3 )  M ( K4 ) , so we can obtain  (M (W3 ))   (M ( K4 ))  8 ,the conclusion is true.
Case2 When n  4 , (M (Wn ))  n  3 from the lemma 5, M (Wn ) exists some vertices of maximum degree
are adjacent,  (M (Wn ))  (M (Wn ))  1  n  4 by lemma 5,7. It is obviously to prove exists n  4 -AVDEC
of M (Wn ) , we only need to construct a map f from E (M (Wn )) to {1, 2,  , n + 3, 0}:
f (u0ui )  i  1, i  1, 2,, n; f (u0v0 )  n;
f (ui v j )  i  j  1(mod n  4), i  1, 2,, n  1; j  1, 2,, n;
f (un v1 )  n; f (un v j )  n  j (mod n  4), i  2,3,, n 1; f (unvn )  n  3
f (vi ui 1 )  n  i  1(mod n  4), i  1, 2,, n 1;
f (u1un )  n  1;
f (wvi )  i, i  1, 2,, n 1; f (wvi )  n  2  i(mod n  4), i  0,1,, n
then we have
C (ui )  {n  i  2}(mod n  4), i  1, 2,, n
Obviously, the f is n  3 -AVDEC of M (Wn ) , the conclusion is true.
From all of above, the conclusion is true.
Theorem 6 For n ≥ 1, then
5, n  1, 2;
n  3, n  3.
 ( M ( Sn ))  
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, 268 - 272, 2016
Proof Let V (Sn )  {u0 , u1 , u2 ,, un },V   {v0 , v1 , v2 ,, vn } ，there are three cases to be discussed as follow:
Case1 When n  1 , M (S1 )  M ( P2 ) , so we can obtain  (M (S1 ))   (M ( P2 ))  5 , the conclusion is true.
Case2 When n  2 , M (S2 )  M ( P3 ) , so we can obtain  (M (S2 ))   (M ( P3 ))  6 , the conclusion is true.
Case3 When n  3 , (M (S2 ))  n  1, by lemma 6, M ( Sn ) exists some vertices of maximum degree are
adjacent,  (M (Sn ))  (M (Sn ))  1  n  2 by lemma 6,7. If exists -AVDEC,then
C (u0 )  C (ui )  C (ui )  1, i  1, 2,, n. (1)
C  u0   C  u1   C  u2   C  un  ;

C  u1   C  v1   C  v2   C  vn  ;

C  u2   C  v1   C  v2   C  vn  ; (2)


C  un   C  v1   C  v2   C  vn  ;

from (1) and (2), we can obtain
C (u0 )  C (u1 )  C (u2 )    C (un )  C (u0 )  C (v1 )  C (v2 )    C (vn )  n  2.
It is obvious that M ( Sn ) don’t exists (n  2) -AVDEC. In order to prove the conclusion be true, we only
need to construct a map f from E (M (Sn )) to {1, 2, • • • , n + 2, 0}:
f (u0ui )  i  1, i  1, 2,, n; f (u0v0 )  n;
f (ui v j )  i  j (mod n  3), i  1, 2,, n; j  1, 2,, n;
f (wvi )  i  1, i  0,1,, n;
then we have
C (u0 )  {n  1, n  2}; C (u1 )  {1, n  2}C (ui )  {i  2, i}1, i  2,3,, n.
C (v1 )  {0, n  2}; C (vi )  {i  2, i  1}, i  2,3,, n.
Obviously, the f is (n  3) -AVDEC of M ( Sn ) , the conclusion is true.
From all of above, the conclusion is true.
From the conclusions of Theorem 1,2,3,4,5,6, the conjecture is true for equal degree graph.
Acknowledgements
This work was supported by Fund Projectsuch as the follow:Hebei Province Natural Science (F2014209086)
2014;the Science and Technology Plan Projects of Tangshan (14130252B,14230248B).
REFERENCES
Balister P N, Bollob´as B, Schelp R H(2002)“Vertex distinguishing coloring of graphs with Δ(G)=2”, Discrete
Mathematics, 252(2), pp.17 29.
Balister P N, Riordan O M and Schelp R H(2003)“Vertex-distinguishing edge coloring of graphs”, J. Graph
Theory ,42，pp.95-109.
Bazgan C Harkat-Benhamdine A, Li H(1999)“On the Vertex-distinguishing Proper Edge-coloring of Graph”, J
Combin Theory Ser B, 75，pp.288-301.
Burris A C and Schelp R H(1997)“Vertex-distinguishing Proper Edge-colorings”, J of Graph Theory, 26(2),
pp.73-82.
Harary F,(1985)“Conditional colorability in graphs”, Graphs and Applications,New York.
J.cerny M.hornak and R(1996）“Sotak Observability of a graph”, Math. Slovaca ,46, pp.21-31.