Applied Mathematics, 2011, 2, 1263-1269 doi:10.4236/am.2011.210176 Published Online October 2011 (http://www.SciRP.org/journal/am)
The Maximum Size of an Edge Cut and Graph Homomorphisms 1
Suohai Fan1, Hongjian Lai2, Ju Zhou3
Department of Mathematics, JiNan University, Guangzhou, China Department of Mathematics, West Virginia University, Morgantown, USA 3 Department of Mathematics, Kutztown University, Kutztown, USA E-mail:
[email protected] Received November 2, 2010; revised July 5, 2011; accepted July 12, 2011
2
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 probabilistic methods that for p ≥ 2 with 1 1 2 2 p 2 if p 0 mod 2 , f p 1 1 if p 1 mod 2 , 2 2 p if there is a graph homomorphism from G onto K p , then b G f p E G . In this paper, we obtained
the best possible lower bounds of b G 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. 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 nonempty 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 for some nonempty proper subset S V G . Define 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 . If there is a graph homomorphism from G to H , then G is called H-colorable. Suppose that G is H-colorable. Then every graph homomorphism :V G V H also induces a map e : E G E H . If for any graph homomorphism Copyright © 2011 SciRes.
:V G V H , and for any e1 , e2 E H , we always have e 1 e1 e 1 e2 , then G is called edge-uniformly H-colorable. Throughout this paper, we use Kp to denote the complete graph with p vertices and C2 p 1 to denote an odd cycle with V C2 p 1 vi , i Z 2 p 1 and N vi vi 1 , vi 1 . Theorem 1.1 (Erdös [2]) Let p ≥ 2 be an integer and 1 1 2 2 p 2 if p 0 mod 2 , f p 1 1 if p 1 mod 2 . 2 2 p
If there is a graph homomorphism from G onto Kp, then b G f p E G . Let p and q be two positive integers with p ≥ 2q. The Kneser graph K p:q is the graph whose vertices repreAM
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sent the q-subsets of 0,1, , p 1 , where two vertices are connected if and only if they correspond to disjoint subsets. Theorem 1.2 Let p , q be two positive integers with p ≥ 2q. If there is a graph homomorphism from G 2q onto K p:q , then b G E G . p Let p and q be two positive integers with p ≥ 2q. The circulant graph K p / q is the graph with vertex set V G 0,1, 2, , p 1 and the neighbors of vertex v are v q, v q 1, , v p q . Theorem 1.3 Let p , q be two positive integers with p ≥ 2q. If there is a graph homomorphism from G 2q onto K p / q , then b G E G . p Theorem 1.4 Let p , q be two positive integers with p ≥ 2q and let p 2 4q 2 4q if p is even, 2 p p 2q 1 f p, q 2 2 p 4q 4q 1 if p is odd. 2 p p 2q 1 If G is an edge-uniformly K p / 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 contains 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 Subsection 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 K p / q -colorable. In Subsection 3.5, we show a best possible lower bound for b G when G has a graph homomorphism onto an odd cycle C2 p 1 and characterize the graphs reaching the lower bound among all edge-maximal C2 p 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 Copyright © 2011 SciRes.
E G to E H such that for each e uv E G , e u v E H . Lemma 2.1 Let g be a function between sets X and Y. Then 1) For any S X , g 1 S g 1 S . 2) For any sets C Y and D Y , C D if and only if g 1 C g 1 D . Lemma 2.2 If D S , S is an edge cut of H, then
1 D 1 S , 1 S is an edge cut of G. Con
1 sequently, D S , 1 S . 1
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 xy 1 D . It follows that 1 D 1 S , 1 S . Conversely, for each xy 1 D , x y xy D S , S . We may assume that x S and y S . Then x 1 S and y 1 S . This proves 1 D 1 S , 1 S . 1 1 1 Therefore, D S , S is an edge cut of G. By Lemma 2.1 1), 1 D 1 S , 1 S . Lemma 2.3 Suppose that D1 , D2 , , Dk is an
edge cut cover of H. Then 1 D1 , 1 D2 , , 1 Dk is an edge cut cover of G. Moreover, if an edge e of H lies in exactly l 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 Di for some Di . It follows that xy 1 Di , and so is an edge cut cover of G. By Lemma 2.1 2), e Di if and only if 1 e 1 Di , and so if an edge e of H lies in exactly l 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 D1 , D2 , , Dk 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 D1 , D2 , , 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 collection D1 , D2 , , Dk of edge cuts of H such that
i 1 Di E H k
and
i 1 Di k
l E H .
Lemma 2.4 Suppose that is an onto graph homomorphism from G to H and D1 , D2 , , Dk is a k k edge cut cover of H such that i 1 1 Di l E G . AM
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l E G . k Proof. By Lemma 2,
Then b G
1 D1 , 1 D2 , , 1 Dk
is an edge cut cover of G and for any edge e of H, if e lies in exactly l -members of , then every edge of 1 e lies in exactly l members of . Therefore, l b G E G follows from k l E G i 1 1 Di kb G . k
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 Theorem 1.1 and determine all graphs G reaching the lower bound in Theorem 1.1 when G is edge-maximal Kp-colorable. p Lemma 3.1 Let p ≥ 2 be an integer and let p , 2 p p k 2 2 l . Then the graph Kp has a k-edge cut p 2 l-cover. p Proof. Let X , V K p \ X : X is a -subset 2 of V K p . Then is an edge cut cover of Kp with p p k . Since every X , V K p \ X 2 p p has size X , V K p \ X , and since 22 Copyright © 2011 SciRes.
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p E K p , every edge of Kp must be in exactly 2 p p k 2 2 l members of . p 2
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 vi V G by Hi, a copy of H, and joining every vertex in Hk to every vertex in Hl if vk vl E G . Theorem 3.2 Suppose that there is an onto graph homomorphism 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 K p K s . Proof. 1) By Theorem 1.1, b G f p E G . Suppose that b G f p E G . Let : V G V K p be an arbitrarily given graph homomorphism and let V K p v1 , v2 , , v p be labelled such that if Vi 1 vi , then V1 V2 V p . For any subset
X V K p , let Y V K p \ X and let also denote the induced map : E G E K p . Then 1 X v X 1 vi v X Vi and i
i
1 Y V G \ 1 X . Since G is an edge-maximal
Kp-colorable, 1 X , Y K
p
1
X , 1 Y V G
is
p a complete bipartite graph. There are k p parti 2 tions of V K p into two parts X i , Yi , i 1, 2, , k , such that 0 Yi X i 1 . Set Di X i , Yi . Label
them so that 1 D1 1 D2 1 Dk . By Lemma 2.2 and Lemma 2.3 and the assumption that l E G b G 1 Dk and b G f p E G , k l k so k E G k 1 Dk i 1 1 Di l E G , k
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p p k 2 2 where l . Therefore all the inequalities are p 2
equalities and so 1 D1 1 D2 1 Dk . Let X v1 , vi1 , vi2 , , vi p with v p X , Y 2 1 V K p \ X and D X , Y . Let X X v p \ v1 , Y V K p \ X and D X , Y . Then 1 D
1 D . Since 1 D 1 X 1 Y V1 Vi1 Vi p 2 1
V p Vi1 Vi p 1 2 V K p V p Vi Vi , 1 p 2 1 it follows from 1 D 1 D that V1 V p , and so V1 V2 V p s for some positive integer 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 G K p K s Conversely, suppose that G K p K s . By Theorem 1.1, b K p s f p K p K s . It remains to show
X ,Y
X , Y f p K p K s .
of V K p K s ,
X ,Y
be an edge cut of K p s such
that 1) X , Y is maximized and subject to 1), 2) i : X i Yi 0 is minimized, where X i X Vi and Yi Y Vi . Then we have the following claims: Claim 1. For each i, 1 i p , X i Yi 0. Copyright © 2011 SciRes.
X ,Y . Claim 2. i : Vi X i : Vi Y 1. Otherwise, i : Vi X i : Vi Y 2 . Choose
trary to the choice of
when p is even and
X ,Y
p
2
1 s
4 that is, X , Y f p E K p K s .
4
2
when p is odd,
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 K p:q has a p-edge cut 2q-cover. Proof. For i 0,1, , p 1 , let Vi X 0,1, 2, , p 1 : X q and i X and p 1 Di Vi , Vl . Then Vi . By the definition of q 1 K p:q , Vi is an independent sets of K p:q and Di is an
Let V1 , V2 , , V p denote the partition of V K p K s such that for i 1, 2, , p, Vi is an independent set of K p K s . Let
Yi 0 1, contrary to the choice of X , Y . Then m t . Without less of generality, we may assume m t . Let X X Yi and Y Y \ Yi . Then X , Y is an edge cut of K p K s such that X , Y X , Y t m Yi X , Y , con-
X , Y X , Y i : Vi X i : Vi Y 1 m2 X ,Y , contrary to the choice of X , Y . p2 s2 By Claim 1 and Claim 2, we have X , Y
1 D 1 X 1 Y
that for any partition
i : X i Yi 0 i : X i
Vi X and set X X \ Vi and Y Y Vi . Then X , Y is an edge cut such that
V K p V1 Vi1 Vi p 2 1
and
Otherwise, there is i such that X i Yi 0. Let m X X i and t Y Yi , If m = t, let X X Yi and Y Y \ Yi Then X , Y is an edge cut of K p K s such that X , Y X , Y and
p 1 p q edge cut of size . Then q 1 q D0 , D1 , , D p 1 is an edge cut cover of K p:q .
p Since K p:q has vertices and each of them is of q p p q p q q q . , degree E K p:q 2 q Since each edge of K p:q is contained in AM
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p 1 p q p q 1 q 2q edges cuts, K p:q has a p-edge p p q q q 2 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 K p:q is induced by the maximum independent set of K p:q and is of size p 1 p q . q 1 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 K p:q 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) K p / q has a p-edge cut at least 2q-cover. 2) K p / q has a p-edge cut average p p 2 q q 1 2 2 -cover. p 1 2 q 1 Proof. For i 0,1, , p 1 , let p Vi i, i 1, , 1 and let Di Vi , Vi . Then 2 D0 , D1 , , D p 1 is an edge cut cover of K p / q with
p p Di q q 1 . 22 1) Notice that for any edge e uv K p / q , e lies in at least 2q members of D0 , D1 , , D p 1 , then K p / q has a p-edge cut at least 2q-cover. p p 1 2 q 1 2) Since E K p , q 2
i 0 Di p
p p q q 1 p 2 2 p p 2 q q 1 22 E K p/q p 1 2 q 1
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and so K p / q has a p-edge cut average p p 2 q q 1 2 2 -cover. p 1 2 q 1
Proof of Theorem 1.3 Theorem 1.3 follows from Theorem 2.5 and Theorem 3.6 1). Proof of Theorem 1.4 Theorem 1.4 follows from Theorem 2.5 and Theorem 3.6 2). Theorem 3.7 Let G be an edge-maximal graph such that G is edge uniformly K p / q -colorable. Then G K p / q K s and b G f p, q E G . Proof. Suppose that G is an edge-maximal graph such that G is edge-uniformly K p / q -colorable. Then, G K p / q K s for some positive integer s. By Theorem
1.4, we have b K p / q K s f p, q E K p / q K s . Now we prove that b K p / q K s f p, q E K p / q K s . For i 0,1,, p 1 V K p / q , let Vi h 1 i , where h : V G V K p / q is an onto graph homomorphism from G to K p / 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 K s such that 1) X , Y is maximized and subject to 1), 2) i : X i Yi 0 is minimized, where X i X Vi and Yi Y Vi . Claim 1. For each i, X i Yi 0 . Proof. Otherwise, there is i such that X i Yi 0 . Since for each vi Vi , either N vi X N vi Y or N vi X N vi Y . If the former is true, then
X \ X i , Y X i X , Y , contradicting the choice of X , Y ; if the latter is true, then X Yi , Y \ Yi X , Y , contradicting the choice of X , Y . j Assume that X i j Vi and let j1 , j2 , , jm J . m
1
p Without loss of generality, we can assume that J . 2 Let C p 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 i j 1 mod p . Let distC p J be the length of the shortest path of C p which contains all the elements in J. Then distC J m 1. Claim 1. distC J m 1. Proof. Suppose, to the contrary, that distC p J m 1. Let P be a path of C p which contains all the elements AM
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in J and assume that j1 , jm are the endpoints of P. Then there is k V P \ J . Let X X \ V jm Vm
and J J \ jm m . Then distC p J distC p J and X , X X , X , a contradiction. p Claim 2. m . 2
2p i 0
Wi 2 p E G 2 p 1 i 1 Wi . 2p
X 0 , we must have W0 W1 W2 p . Let m W0 .
3.4. Graphs with a Graph Homomorphism onto an Odd Cycle In this section, we will show a best possible lower bound for b G when G has a graph homomorphism onto an odd cycle C2 p 1 and characterize the graphs reaching the lower bound among all edge-maximal C2 p 1 -colorable graphs. Theorem 3.8 Suppose that there is an onto graph homomorphism from G to C2 p 1 . Then each of the following holds. 2p E G , where equality holds only 1) b G 2 p 1 if G is edge-uniformly C2 p 1 -colorable. 2) Among all edge-maximal C2 p 1 -colorable graph G, 2p b G E G if and only if G C2 p 1 K s . 2 p 1 Proof. 1) Notice that C2 p 1 K (2 p 1)/(2 p ) , 2p b G E G follows from Theorem 1.3. Now 2 p 1 2p E G . Let suppose that b G 2 p 1 : V G V C2 p 1 be an arbitrarily given graph homomorphism and for each i Z 2 p 1 , let Vi 1 vi . Set Wi Vi , Vi 1 , for each i Z 2 p 1 . Since is a graph homomorphism, we have E G iZ Wi . It 2 p 1
follows that E G i 0 Wi . We may assume that 2p
W0 min Wi : i Z 2 p 1 . Then E G W0 is an edge-
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lows that 2 p
Hence 2 p W0 W1 W1 W2 p . By the choice of
p Proof. Suppose, to the contrary, that m . Let P 2 be a path of C p which contains all the elements in J and assume that j1 , jm are the endpoints of P. Let jm 1 V C p \ V P be such that jm jm 1 E C p . Let X X Vm 1 and J J jm 1 . Then p distC J distC J 1 and X , X X , X , 2 a contradiction. By the above discussion, we can calculate that b K p / q K s f p, q E K p / q s K s .
cut of G , and so by the assumption on G , we have 2p 2p E G b G E G W0 i 1 Wi . It fol2 p 1
Then for any edge e E C2 p 1 , 1 e m . Since
is arbitrarily, then G is edge-uniformly C2 p 1 -colorable.
2) Suppose that G is an edge-maximal C2 p 1 -color2p E G . Let Wi be deable graph with b G 2 p 1 fined as in 1). Since G is an edge-maximal C2 p 1 -colorable graph, the subgraph induces by Wi is a complete bipartite graph. Since 2p E G , by 1), G is edge-uniformly C2 p 1 b G 2 p 1 colorable, which means, W0 W1 W2 p Therefore, V0 V1 V2 p s for some positive integer s and so G C2 p 1 K s . Conversely, suppose G C2 p 1 K s . Note that
E C2 p 1 K s 2 p 1 m 2 . Then
b C2 p 1 K s 2 pm 2 and so it suffices to show that for
any subset X V C2 p 1 K s
and
Y V C2 p 1 K s \ X , the edge cut X ,Y of C2 p 1 K s
X , Y 2 pm 2 . Let X , Y be an edge cut of C2 p 1 K s 1) X , Y is maximized and subject to 1), 2) satisfies
i : X i
such that
Yi 0 is minimized, where X i X Vi and
Yi Y Vi . Since iI Vi , iI Vi , where 1 2 I1 i Z 2 p 1 : i is odd and I 2 i Z 2 p 1 : i is even , is an edge cut with cardinality 2 pm2, then
X , Y iI Vi , iI 1
2
Vi 2 pm 2 . Then we have the
following claims: Claim 1. For each i, X i Yi 0. Proof. Otherwise, there is i such that X i Yi 0 . Let s X i 1 X i 1 and t Yi 1 Yi 1 . If s t , let X X Yi and Y Y \ Yi . Then X , Y is an edge cut of Gm such that X , Y X , Y and AM
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$ i : X i Yi 0 i : X i Y 0i 1 , contrary to the
choice of X , Y . Then s t . Without loss of generality, we may assume s < t. Let X X Yi and Y Y \ Yi. Then X , Y is an edge cut of Gm such that X , Y X , Y t s Yi X , Y , contrary to the
choice of X , Y . Claim 2. There is exactly one pair of consecutive number i, i 1 in 0,1, , 2 p (we consider 0 and 2p as a pair of consecutive numbers, too) such that X i X i 1 . Proof. By Claim 1 and the structure of C2 p 1 K s , there exist some pair of consecutive numbers i, i 1 in
such that X i X i 1 Since X , Y 2 pm 2, then there is at most one pair of consecutive number i, i 1 in 0,1, , 2 p such that X i X i 1 and so Claim 2 follows. By Claim 1 and Claim 2, X , Y 2 pm 2.
0,1,, 2 p
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A. J. Bondy and U. S. R. Murty, “Graph Theory with
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Applications,” American Elsevier, New York, 1976. [2]
P. Erdös, “Problems and Results in Graph Theory and Combinatorial Analysis,” Graph Theory and RelatedTopics, Academic Press, New York, 1979.
[3]
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M. O. Albertson, P. A. Catlin and L. Gibbons, “Homomorphisms of 3-Chromatic Graphs II,” Congressus Numerantium, Vol. 47, 1985, pp. 19-28.
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