The Maximum Size of an Edge Cut and Graph Homomorphisms

(1)

The Maximum Size of an Edge Cut and Graph

Homomorphisms

Suohai Fan1, Hongjian Lai2, Ju Zhou3

1

Department of Mathematics, JiNan University, Guangzhou, China 2

Department of Mathematics, West Virginia University, Morgantown, USA 3

Department of Mathematics, Kutztown University, Kutztown, USA E-mail: zhou@kutztown.edu

Received November 2, 2010; revised July 5, 2011; accepted July 12, 2011

Abstract

For a graph G, let b G

 

max



D D: is an edge cut of G



. For graphs G and H, a map



:V G

 

V H

 

is a graph homomorphism if for each euvE G

 

,



   

u



v E H

 

. In 1979, Erdös proved by

prob-abilistic methods that for p ≥ 2 with

 









1 1

if 0 mod 2 ,

2 2 2

1 1

if 1 mod 2 , 2 2

p p

f p

p p

  

 _

  

  



if there is a graph homomorphism from G onto Kp, then b G

 

 f p E G

   

. In this paper, we obtained

the best possible lower bounds of for graphs G with a graph homomorphism onto a Kneser graph or a

circulant graph and we characterized the graphs G reaching the lower bounds when G is an edge maximal

graph with a graph homomorphism onto a complete graph, or onto an odd cycle.

 

b G

Keywords:Maximum Edge Cuts, Graph Homomorphisms

1. Introduction

In this paper, the graphs we consider are finite, simple and connected. Undefined notation and terminology will follow those in [1]. Let G be a graph. For disjoint non-empty subsets X Y, V G

 

, let



X Y,



denote the set

of edges of G with one end in X and the other end in Y. For a subset S V G

 

, let S V G

 

\ .S An edge cut

of G is an edge subset of the form S S,



G 

for some nonempty proper subset SV



. Define

 

max



t of G



b G  D D: is an edge cu .

For graphs G and H, a map :V G

 

V H

 

is a

graph homomorphism if for each euvE

 

G ,

. If there is a graph homomorphism from

   

to

 

u v E H

  

G H , then is called H-colorable. Sup-pose that is H-colorable. Then every graph

homo-morphism also induces a map

If for any graph homomorphism

G

V G

:V G



 



:

e E G E H

 

 



. 

 

H

 

:V G V H

  , and for any e1, e2E H

 

, we always have 1

 

1

 

2 1

e e

 _ 

e e , then G is called

edge-uniformly H-colorable.

Throughout this paper, we use Kp to denote the

com-plete graph with p vertices and C2p1 to denote an odd cycle with V C



2p1

 

 v ii, Z2p1



and

  

_i v_i ₁,v_i ₁



.

N v  _ _

Theorem 1.1 (Erdös [2]) Let p≥ 2 be an integer and

 









mod 2 ,

mod 2 . 1 1

if 0 2 2 2

1 1

if 1 2 2

p p

f p

p p

  

 _

  

 _ _



If there is a graph homomorphism from G onto Kp,

then b G

 

 f p E G

   

.

(2)

repre-

sent the q-subsets of where two vertices are connected if and only if they correspond to disjoint subsets.



0,1, , p1 ,

p q

Theorem 1.2 Let , be two positive integers

with p≥ 2q. If there is a graph homomorphism from G onto Kp q_: , then

 

2q _.

b G E G

p



Let p and q be two positive integers with p≥ 2q. The circulant graph Kp q/ is the graph with vertex set

and the neighbors of vertex v are

  



0,1, 2,

V G  

vq vq v

p

,p 1 , 1,



.



,  p q



q

with p≥ 2q. If there is a graph homomorphism from G onto Kp q/ , then

 

2

. q

b G E G

p



with p≥ 2q and let

p q













2 2

4 4 _if _{is even,} 2 2 1

,

4 4 1

if is odd. 2 2 1

p q q

p

p p q

f p q

p q q

p

p p q

  

 _ _

  

   

 _ _



If G is an edge-uniformly Kp q/ -colorable, then

 



,

  

.

b G  f p q E G

A graph G is edge-maximal H-colorable if G is H-colorable but for any graph G such that G con-tains G as a spanning subgraph with E G

 

  E G

 

,

G is not H-colorable.

In Section 2, we prove an associate Theorem which will be used in the proofs of other theorems. In Subsec-tion 3.1, we give an alternative proof of Theorem 1.1 and characterize the graphs G reaching the lower bound in Theorem 1.1 when G is edge-maximal Kp-colorable. In

Subsection 3.2, we show the validity and sharpness of Theorem 1.2. In Subsection 3.3, we show the validity of Theorem 1.3 and Theorem 1.4 and characterize the graphs G reaching the lower bound in Theorem 1.4 when G is edge-maximal Kp q/ -colorable. In Subsection 3.5, we show a best possible lower bound for when G has a graph homomorphism onto an odd cycle 2 1

 

b G

p

C 

and characterize the graphs reaching the lower bound among all edge-maximal C2p1-colorable graphs. There are a lot of researches about graph homomorphism can be found in [3-7].

2. An Associate Theorem

In this section, we shall prove an associate Theorem which will be used in the proofs of other theorems. Throughout this section, we assume that G and H are two graphs and  is an onto graph homomorphism from G to H. Note that we can also view  as a map from

 

E G to E H

 

such that for each euvE G

 

,

 

e

   

u v E H

 

 

S X



 



.

Lemma 2.1 Let g be a function between sets X and Y. Then

1) For any , 1

 

1

 

S g S g .

2) For any sets and , if and

only if

CY DY CD

 

1 1 _.

g C g D

Lemma 2.2 If D S S, _ is an edge cut of H, then

 

1 1 _,

D S

  _1

S 

   is an edge cut of G. Con- sequently, 1

 

1

 

_, 1

 

_.

D S S

 _  _

  

Proof. For each x1

 

S , y1

 

S with

 

xyE G , by the definition of inverse image, 

 

x S,

and 

 

y S. Hence, 

 

xy 

   

x  y D, and so

 

1

xy D . It follows that  1

 

D   1

 

S ,1

 

S 

 1

  _.

Conversely, for each xy

 

D ,

   

x y 

 

xy D S S,

   _{  } _. We may assume that

 

x S

  and 

 

y S . Then x1

 

S and

 

S

1

y . This proves 1

 

D  1

 

S ,1

 

S _.

Therefore, _1

 

_D __1

 

_S _,_1

 

_S 

  is an edge cut of G. By Lemma 2.1 1), 1

 

D  1

 

S ,1

 

S _.

Lemma 2.3 Suppose that is an

edge cut cover of H. Then



D D1, 2, ,Dk



 



 



1



1 , 2 , , k

D D  D

1 1

 

  

l

is an edge cut cover of G. Moreover, if an edge e of H lies in exactly

 members of , then every edge in 1

 

e

 _{lies in}

exactly l members of .

Proof. By the definition of graph homomorphism, for each exyE

 

G , 

 

xy 

   

x  y E H

 

. Since

is an edge cut cover of H, then 

 

xy

   

x  y i

  D for some _i . It follows

that

D 

 

1 _,

i

xy D and so is an edge cut cover of

G. By Lemma 2.1 2), e i if and only if

 

D

 

1

 

i



1

e  D

 l

, and so if an edge e of H lies in ex-actly  members of  , then every edge of 1

 

e



lies in exactly l members of .

Let k, l be two positive integers. A k-edge cut at least l-cover of a graph H is a collection

of k edge cuts of H such that every edge of H lies in at least l members of . A k-edge cut l-cover of a graph H is a collection



D D1, 2, ,Dk



 





1 2 ,Dk



of k edge cuts of

H such that every edge of H lies in exactly l members of . A k-edge cut average l-cover of a graph H is a col-lection

, ,

D D 

 





D D1, 2,,Dk





 

H

 of edge cuts of H such that

1

k i i



D E and ₁ _i

 

i D l E H



k

.

Lemma 2.4 Suppose that  is an onto graph homo-morphism from G to H and 



D D1, 2, , Dk



is a k

 



edge cut cover of H such that ki₁ 1 Di l E

 

 _

(3)

Then b G

 

l E G

 

. k



Proof. By Lemma 2,

 



1 1 1



1 , 2 , , k

D D D

  

  



is an edge cut cover of G and for any edge e of H, if e lies in exactly -members of , then every edge of

lies in exactly members of l . Therefore,

 

 

1

e



l 

 

l

 

b G E G

k

 follows from

 

1

 

1 .

k

i i

l E G 



_  D kb G

Theorem 2.5 If one of the following is true, then

 

l

 

.

b G E G

k



1) There is a graph homomorphism from G onto H and H has a k-edge cut l-cover.

2) There is a graph homomorphism from G onto H and H has a k-edge cut at least l-cover.

3) G is edge-uniformly H-colorable and H has a k-edge cut average l-cover.

3. Main Results

3.1. Graphs with a Graph Homomorphism onto a Complete Graph

In this section, we present an alternative proof for Theo-rem 1.1 and determine all graphs G reaching the lower bound in Theorem 1.1 when G is edge-maximal Kp-colorable.

Lemma 3.1 Let p≥ 2 be an integer and let

2

p

      _ _  _{ }   ,

2 2

2 p p k l

p

            

     

. Then the graph Kp has a k-edge cut

l-cover.

Proof. Let ,





: is

2 \

p

X V K X X

_ _  

 _ _ _{ }

  

 a -subset

of V K

 

p . Then  is an edge cut cover of Kp with

  

. k



 







2 p

p

      _ _  _{ }  

Since every X V K, p\X 

has size _X V K,



_p\X



   _  p

 2 2

p

   

 , and since

 

, 2

p

p E K    

  every edge of Kp must be in exactly

2 2

2 p p k

l p

            

     

members of  .

Thus, Theorem 1.3 follows from Theorem 2.5 1) and Lemma 3.1.

The lower bound of Theorem 1.1 is best possible, in the sense that there exists a family of graphs such that the lower bound of Theorem 1.1 is reached.

Let G and H be two graphs. The composition of G and H, denoted by G H

 

, is the graph obtained from G by replacing each vertex of by Hi, a copy of H,

and joining every vertex in Hk to every vertex in Hl if

 

i G

v V

 

.

k l

v v E G

Theorem3.2 Suppose that there is an onto graph ho-momorphism from G to Kp and G is edge maximal

Kp-colorable.

Then each of the following holds.

1) b G

 

 f p E G

   

, where equality holds only if G is edge-uniformly Kp-colorable.

2) Among all edge-maximal Kp-colorable graph G,

 

   

b G  f p E G if and only if GKp  Ks .

Proof. 1) By Theorem 1.1, b G

 

 f p E G

   

.

Sup-pose that b G

 

 f p E G

   

. Let :V G

 

V K

 

p

be an arbitrarily given graph homomorphism and let

  

p 1, , ,2 p



V K  v v  v be labelled such that if

 

1

i

V  vi , then V1 V2  Vp . For any subset

 

p ,

X V K let Y V K

 

p \X and let  also

de-note the induced map :E G

 

E K

 

p . Then

 

1 1

i i i i

v X v X

X v V



_

 _

_

  _  _ and

 

Y V G

 

\

 

X

  





1 1

  _{. Since}

G is an edge-maximal Kp-colorable, 1



,



1

 

, 1

 

_{ }

p

K X Y V G

 _  _

  

X Y is

a complete bipartite graph. There are

2

p

k p

      _ _  _{ }  

parti-

tions of V K

 

p into two parts



X Yi, i



,i1, 2, , k,

such that 0Y_i  X_i 1. Set D_i 



X Y_i, _i



. Label

them so that 1

 

1

 

1

 

1 2 k .

D D D

 _ ___ _ _By

Lemma 2.2 and Lemma 2.3 and the assumption that

 

   

,

b G  f p E G

 

1

 

k

l

E G b G D

k 



  and

so

 

1

 

1

 

1

k

k i i

l

k E G k D D l E G

k  

 



(4)

where 2 2

2

p p

k l

p

            

     

. Therefore all the inequalities are

equalities and so 1

 

1

 

1

 

1 2 k .

D D D

 _  ___

Let ₁ ₂

1 2 1, ,i i , , i_p

X v v v v

     

 



   

 

 

 _ with v_pX, Y 

 

_p \

V K X and D



X Y, 



. Let XX

 

vp \

 

v1 , and

 

p

Y V K \X D 



X ,Y



. Then 1

 

D 

 

1 _D

 _ _{. Since}

 

1

1 2

1

1 2

1 1 1

1

p

i i

p i i

D X Y

V V V

V K V V V

  

     

       _ _  _  _

 

 

   

 

 

  

  

    

 

 _ _

 



and

 

1

1 2

1

1 2

1 1 1

, p

p

p i i

p p i i

D X Y

V V V

V K V V V

  

     

       _ _  _  _

 

 

   

 

 

  

     

 

 _ 





 it follows from _1

 

_D_ __1

 

_D__ _that

1 p

V V ,

and so V1 V2 Vp s for some positive

inte-ger s, which implies G is edge uniformly Kp-colorable.

2) Suppose that G is an edge-maximal Kp-colorable

graph such that b G

 

 f p E G

   

. By the proof of

1), there is some positive integer s such that

s p

GK _K _

Conversely, suppose that GK_p_Ks_. By Theorem

1.1, b K



p

 

s



 f p K

 

pKs. It remains to show that for any partition



X Y,



of V



K_p_Ks_



,



X Y,



 f

 

p KpKs.

Let V V1, , ,2Vp denote the partition of V



KpKs



such that for i1, 2, , , p V_i is an independent set of

s p

K  K . Let



X Y,



be an edge cut of K_p

 

s such

that 1)



X Y,



is maximized and subject to 1), 2)



i X Y: i i 0 is minimized, where



Xi XVi and i

Y YV_i. Then we have the following claims:

Claim 1. For each i, 1 i p, X Y_i _i 0.

Otherwise, there is i such that X Yi i 0. Let

i

m X  X and tY Yi , If m = t, let X XYi

and Y Y Y\ i Then



X Y ,



is an edge cut of KpKs

such that



X Y,

 

 X Y ,



and



i X Y: i i 0

 

 i X Y: i i 0



1, contrary to the choice of



X Y,



. Then . Without less of

general-ity, we may assume

mt t

m . Let X  XY_i and

. \ i

Y Y Y Then



X Y ,



is an edge cut of K_pKs

such that



X  ,Y

 

X Y,



 



t m Y



_i 



X Y,



,

con-trary to the choice of



X Y,



.

Claim 2.



i V: iX

 

 i V: iY



1.

Otherwise,



i V: iX

 

 i V: iY



2. Choose

and set

i

V X X X V\ _i and Y YV_i. Then



X Y ,



is an edge cut such that



 







 









2

, , : : 1

, ,

i i

X Y X Y i V X i V Y

X Y

       



m

contrary to the choice of



X Y,



.

By Claim 1 and Claim 2, we have



,



2 2 4

p s

X Y 

when p is even and









2 ₁ 2

,

4

p s

X Y   when p is odd,

that is,



X Y,



 f p E K

 



_pKs



.

3.2. Graphs with a Graph Homomorphism onto a Kneser Graph

In this section, we shall show the validity and sharpness of Theorem 1.2.

Theorem 3.3 Let p and q be two positive integers with p≥ 2q. Then Kp q: has a p-edge cut 2q-cover.

Proof. For i0,1, , p1, let







0,1, 2, , 1 : and



i

V  X   p X q iX and

, l

i i

D  V V_. Then 1 .

1

i

p V

q

     _

  By the definition of

:

p q

K , V_i is an independent sets of K_{p q}_: and is an

edge cut of size

i

D

1 1

p p

q q

q

 

  _  _     

 

. Then



D D0, 1, ,Dp1



 

 is an edge cut cover of Kp q: .

Since Kp q: has vertices and each of them is of

degree

p q

     

p q q



 

 

 ,

 

p q: 2 .

p p q

q q

E K

          

(5)

1 1

2

p p q

p

q q

q

p p q

q q

 

  _ _  _        _

    

        

edges cuts, Kp:q has a p-edge

cut 2q-cover.

Proof of Theorem 1.2 Theorem 1.2 follows from

Theorem 2.5 and Theorem 3.3.

Theorem 3.4 (Poljak and Tuza, Theorem 2 in [8]). If p≤ 3q, then the maximum edge cut of Kp q: is induced by the maximum independent set of Kp q: and is of size

1

. 1

p p

q q

 

     _     

q

 

Theorem 3.5 The bound in Theorem 1.2 is best possible. Proof. By Theorem 3.4, the edge cuts we choose in the proof of Theorem 4.1 are the maximum edge cuts of

:

p q

K when p≤ 3q. Therefore, the bound in Theorem 1.2 is reached by K_{p q}_: when p≤ 3q.

3.3. Graphs with a Graph Homomorphism onto a Circulant Graph

In this section, we verify the validity and sharpness of Theorem 1.3 and Theorem 1.4.

Theorem 3.6 Let p and q be two positive integers with p≥ 2q. Then the following hold.

1) Kp q/ has a p-edge cut at least 2q-cover. 2) Kp q/ has a p-edge cut average









2 1

2 2

1 2 1

p p q q

p q

_{    }      _{   } 

 

   -cover.

Proof. For i0,1, , p1, let

, 1, , 1

2

i

p V i i  _{ } 

 

   and let Di  V Vi, i



. Then

is an edge cut cover of



D D0, 1, ,Dp1

 

 K_{p q}_/ with



1 .



2 2

i

p p D    _{   }q

    q

1) Notice that for any edge euvKp q_/



D D, , , D

, e lies in at least 2q members of   0 1 p1



, then Kp q/ has a p-edge cut at least 2q-cover.

2) Since



1 2



1



2

p q

p p q

E K

    

    

  ,

















0

/ 1

2 2

2 1

2 2

1 2 1

p i i

p q

p p

D q q p

p p q q

E K

p q



     __{   }  _

   

 

_{    } _     _{   } 

 



  



and so Kp q/ has a p-edge cut average









2 1

2 2

1 2 1

p p q q

p q

_{    } _     _{   } 

 

   -cover.

Theorem 2.5 and Theorem 3.6 1).

Theorem 2.5 and Theorem 3.6 2).

Theorem 3.7 Let be an edge-maximal graph such that is edge uniformly

G

G Kp q/ -colorable. Then

/ s

p q K

GK _ _ and b G

 

 f p q E G



,

  

.

Proof. Suppose that G is an edge-maximal graph such that G is edge-uniformly Kp q/ -colorable. Then,

/ s

p q K

GK _ _ for some positive integer s. By Theorem 1.4, we have b K



_p_/_qKs



 f p



,q E K





_p_/_qKs



.

Now we prove that



_p/_q s





,





_p/ s



b K K _  f p q E K q_K  . For i



0,1, , p 1



V K



p q/



, let

 

1

i

V h i, where

 



/



: G V Kp q

h V is an onto graph homomorphism

from G to Kp q/ such that for any edge xy of G,

   

q h x h y  p q. By the definition of graph homomorphism, Vi is an independent set. Let



X Y,



be an edge cut of K_{p q}/ Ks such that 1)



X Y,



is

maximized and subject to 1), 2)



i: X Y_i _i 0



is mi-nimized, where XiXVi and YiYVi.

Claim 1. For each i, X Yi i 0.

Proof. Otherwise, there is i such that X Yi i 0.

Since for each viVi, either N v

 

i X  N v

 

i Y

or N v

 

i X  N v

 

i Y . If the former is true, then



X\Xi, YXi

 

 X Y,



, contradicting the choice of



X,Y



; if the latter is true, then



XY Y Y_i, \ _i

 

 X Y,



,

contradicting the choice of



X Y,



. Assume that

1

m j

i i j

X 



_ V and let



j1, , ,j2  jm



J.

2

p J    

 . Without loss of generality, we can assume that

Let Cp be the cycle with vertex set



0,1,2,,p 1



V K



p q/



, where i is adjacent to j if

and only if



p



. Let

 

p C

dist J be the 1

i j mod hortest path of

length of the s C_p which contains all the elements in J. Then distC

 

J 1.

Claim 1.

m



 

1

C

dist J  m .

ppose, to t

Proof. Su the contrar

 

1

p

C

dist J  m

y, tha .

(6)

in J and assume that j1, jm are the endpoints of P. Then there is kV



\ .J Let



\

 



 

m



P X  X Vj  Vm

and J J\

   

jm  m . Then distCp

 

J distCp

 

J

and X X, X, rad

Cl

X

     

    , a cont iction.

aim 2. .

2

p m   

 

Proof. Suppose, to the contrary, that . 2 p m   

  Let P

be a path of Cp which contains all the J

d assume that j1, jm are the endpoi P. Let

elements in

an nts of

 

1 \

m p

j  V C V P be such that j jm m1E C

 

p . Let



m 1



X X V _ and J J

 

j_m_₁ . Then

 

_{   }1  p and 2

 

C C

dist J dist J _X X , _  _X X, , a contradiction.

By the above discussion, we an calculate that c



/









/

 



raph

, _{p q}

p q E K s .

p q s s

b K K _  f _K _

3.4. Graphs with a Graph Homomorphism onto an Odd Cycle

for s a g homomorphism onto an

dd

In this section, we will show a best possible lower bound

 

b G when G ha

o cycle C₂p₁ and characterize the graphs reaching

the lower bound among all edge-maximal 2p1

C orable graphs.

Theorem Suppose that there is an onto graph ho-momorphism from G to 2 1

 -col

3.8

p

C  . Then each of the

fol-lowing holds.

1)

 

2

 

,

2 1

p

b G E G

p



 where equality holds only

if G is edge-uniformly C2p1-colorable. ong all edge-maxim

2) Am al C2p1-colorable graph G,

 

2p

 

if an

2 1

b G E G

p



 d only if GC2_p_1Ks.

Proof. 1) Notice that C2p1K(2p1)/(2 )p ,

 

2

 

2 1

p

b G E G

p



 follows from Theorem 1.3. N w o

suppose that b G

 

 2

 

2p1

momorphism ₁

p

E G . Let

be an arbitrarily given graph

ho- 



2 1



:V G V C _p

  _

and for each iZ₂_p_ , let V_i1

 

v_i .

Set W_i 



V_i,V_i_₁



, for each iZ₂_p_₁. Since  is a

at

graph homomorphism, we have

 

2p1 i

i Z

E G W

 



_

. It

follows th

 

2p₀ i i

E G 



_ W . We may assume that





0 min i : 2p 1

W  W iZ _ . Then -

tion on we h

 

0

E G W is an edge

cut of G, and so by the assump G, ave

 

2

0 1

2 1

p i i

E G W W

p 



 . It

fol-2p

b G E G

  

lows that 2



2p



2

  

2



2p

i i

0 1 1

i i

p



_ W  p E G  p



_ W .

Hence 2p W0 W1 W1   W2p . By the choice of

0

X , we must have W₀ W₁ W₂_p . Let m W0 . Then for any edge eE C



2p1



,

 

1

e m

 _ _{. Since}

 is ar mly

1

bitrarily, then G is edge-unifor

2p

C _ -colorable.

2) Suppose that G is an edge-maximal C2p1-color- le graph with

 

ab 2

 

2 1

b G E G

p



 . Let

as in 1). Sin

p

be de-

ce G is an edge-maximal

i

W

fined 2p1

C _ -colorable graph, the subgraph induces by W_i is a complete bipartite

 

graph. Since

 

2p

2p1E G , by 1), G is edge-uniformly



b G C₂p₁-

colorable, which means, W₀ W₁ W₂p

here-fore,

T

0 1 2p

V V V s for some positive integer o

s and s GC2_p_1_Ks_.

Conversely, suppose GC2_p_1Ks. Note that







₂ _{1 m .}



2

s p

K

     

2p1

E C Then



_ _



2 _{and so it suffices to show t} ubset

2p1Ks 2 pm hat for

b C

any s  X V C



2_p_1Ks



and



C2p1 Ks



\

Y V _ _ _ X, the edge cut



X Y,



of C₂p₁Ks

satisfies



_{X Y}_,



__{2 pm}2_.

Let



X Y,



be an edge cut of C₂_p_₁Ks such that

1)



X Y,



is maximized and subject to 1), 2)



i X Y: _i i 0



is mini zed, where mi Xi XVi and .

i i

Y YV ince

1Vi, 2Vi

 

S _



_{i I}



_{i I} _, where





1 2p1: is odd

I    i and I2 



i Z2p1: ii s even ,



n

i Z

is an edge cut with cardinality 2 pm2_{, the}







i I₁Vi



2 Then we

2

, , i I i 2 pm .

X Y  _ _ V_  have the

following claims:

Claim 1. For each i, X Yi i 0.

Proof. Otherwise, there is i such that X Yi i 0. Let

1 1

i i

s X  X and tYi1 Yi1 . If st, let

i

X XY and Y Y Y\ i. Then



X Y ,



is an edge

(7)

$



i: X Yi i 0

 

 i: X Yi 0i 1, contrary to the choice of



X Y,



. Then st.

. Le

With oss o ity, we ma s < t

out l f general-y assume t XXYi and YY Y\ i.

Then



X Y, 



is an edge cut of G_m such that



X Y, 

 

 X,Y



 



t s Y



i 



X,Y



, contrary to the

choice of



X Y,



.

Claim 2. There is exactly of con

numbe

one pair secutive r



i i, 1



pair of c

in



0,1, , 2 p



(we consider 0 and

2p as a onsecutive numbers, too) such that 1

i i

X  X_ .

Proof. By Claim 1 and the structure of C2p1Ks, there exist som pair ofe consecutive numbers



i i, 1



in



0,1, , 2 p



such that Xi  Xi1 Since





2

, 2 pm

X Y  , then there is at most one pair of consecutive number



i,i1



in



0,1, , 2 p



such that Xi  Xi1 and so

By Claim 2,

Claim 2 follows.

1 and Claim



_,



__{2 pm}2

X Y .

4. References

[1] . Murty, “Graph Theory

Applications,” American Elsevier, New York, 1976.

ted-s, Vol. 54, No. [2] P. Erdös, “Problems and Results in Graph Theory and

Combinatorial Analysis,” Graph Theory and Rela Topics, Academic Press, New York, 1979.

[3] M. O. Albertson and K. L. Gibbons, “Homomorphisms of 3-Chromatic Graphs,” Discrete Mathematic

2, 1985, pp. 127-132.

doi:10.1016/0012-365X(85)90073-1

[4] M. O. Albertson, P. A. Catlin and L. Gibbons, “Homo-morphisms of 3-Chromatic Graphs II,” Congressus

Nu-ies B, Vol. 45, No. merantium, Vol. 47, 1985, pp. 19-28.

[5] P. A. Catlin, “Graph Homomorphisms onto the 5-Cycle,” Journal of Combinatorial Theory, Ser

2, 1988, pp. 199-211.

doi:10.1016/0095-8956(88)90069-X

[6] E. R. Scheinerman and

Theory,” John Wiley Sons, Inc., New

D. H. Ullman, “Fractional Graph York, 1997.

l. 153,

Combinatorics, Vol. 3, [7] G. G. Gao and X. X. Zhu, “Star-Extremal Graphs and the

Lexicographic Product,” Discrete Mathematics, Vo 1996, pp. 147-156.

[8] S. Poljak and Z. Tuza, “Maximum Bipartite Subgraphs of Kneser Graphs,” Graphs and

1987, pp. 191-199. doi:10.1007/BF01788540