graphs with forbidden homomorphic images

PDFlib PLOP: PDF Linearization, Optimization, Protection

Page inserted by evaluation versionwww.pdflib.com – [email protected]

GRAPHS WITH FORBIDDEN HOMOMORPHIC IMAGES

P. Hell*

Department of Mathematics Rutgers University

New Brunswick, New Jersey 08903

D. J. Miller

Department of Mathematics University of Victoria

Victoria, British Columbia Canada V8W 2Y2

In this note we prove the corrected version of the results announced in [7] and [S] describing all graphs H for which the property of having H as a homomorphic image is “almost equivalent” to the property of having chromatic number less than or equal to chr(H) (see below for the precise statement).

Let G, H be graphs. A homomorphism f : G + H is a vertex-mapping V(G) -+ V(H) which preserves adjacency, i.e., such that [g, g’] E E(G) implies [ f (g ) , f (g’)] E E(H). Note that a homomorphism G -+ K, is just an n-coloring of G. Thus the chromatic number of G, chr(G), is the smallest n such that G admits a homomorphism G -+ K , . The existence of a homomorphism G -+ H implies that the chromatic number of G is less than or equal to chr(H), since, for any n-coloring of H, H + K,, the composition G + H + K, is an n-coloring of G (cf. [ 11, p. 144). On the other hand, if the length of a shortest odd cycle in H is greater than the length of a shortest odd cycle in G, then there is no homomorphism G + H even if chr(G) I chr(H). For instance, there is no homomorphism C , -+ C 5 , although chr(C3) = chr(C5) = 3.

Letf : G + H be a homomorphism. Ifboth f : V(G) + V(H)andfX : E(G) -+ E(H) (defined by f”([g, g‘]) = [f(g), f (g’)] for [g, g‘] E E(G)) are surjective (i.e., onto) mappings, we say that H is a homomorphic image of G. A congruence on G is an equivalence relation on V(G) for which each equivalence class (henceforth called a congruence class) is an independent set. Let C be a congruence on G. The quotient G/C is defined as the graph whose vertices are the congruence classes of C, and in which two classes are adjacent if and only if some vertex of one class is adjacent in G to some vertex of the other class. Note that the mapping that assigns to each vertex of G the congruence class to which it belongs, is a homomorphism G -+ G/C and G/C is a homomorphic image of G. Similarly, if H is a homomorphic image of G under f , then H z G/C, where C = ((9, 9‘) : f ( g ) = f (g’)}. (If g is equivalent to g’ under the congruence C, we write (9, g’) E C.) In other words, quotients and homomorphic images coincide. We shall abbreviate the statement “H is a homomorphic image (quotient) of G” by H s G and its negation by H$ G. The relation 9 i. reflective and transitive in the class te of all graphs; in the class of all finite graphs, this relation is also antisymmetric, i.e., it is a partial order, (FIGURE 1).

Let N denote the natural numbers. Note that chr: ($,A ) -+ (N, I) is a decreasing mapping, i.e., that HI G implies chr(H) 2 chr(G). This follows from the remarks made earlier: any n-coloring of H = G/C gives rise to an n-coloring of G by coloring

Partially supported by National Science Foundation grant MCS 77-04949.

270

0077-8923/79/0319-0270 $01.7511 0 1979, NYAS

Hell & Miller: Graphs with Forbidden Homomorphic Images 271

FIGURE 1. An example where both G g H and H s G hold.

each vertex of G with the color of the congruence class of C to which it belongs. We shall refer to this coloring of G as being induced by the coloring of H.

The reducing congruence R on a graph G is defined as

R = { ( g , 9’) : [a XI E W)-- [g’, XI E E(G) )

(It is obvious that R is a congruence.) The quotient G/R is called the reduced graph of G, or the type of G. If G / R E G / R , we shall say that G and G are ofthe same type. A graph is irreducible if the reducing congruence has only one-element classes, i.e., if distinct vertices have distinct neighborhoods. Note that any reduced graph is irreducible.

Let H be a fixed graph. Consider all graphs G such that H $ G, i.e., the class of all graphs for which H is a forbidden homomorphic image. Since this class contains all graphs with chromatic number greater than ch@), a more interesting class is

F(H) = {G E Y : chr(G) s chr(H) and H$ G}. Thus F(H) is the class of all counterexamples G to the statement

HS Gochr(G) I chr(H).

It is to be expected, as is indeed the case, that F(H) is very large for most graphs H. However, we proved in [7] that F(K, ) contains only graphs of finitely many types. (Equivalently, we proved that there are only finitely many types of graphs with a given achromatic number [8].) To simplify the statements, we let Z(H) denote the class consisting of all the types of the graphs of F(H), i.e.,

Z(H) = {G E Y : G irreducible, chr(G) I chr(H), and H$ G}.

Hence Z(K,) is finite (except, of course, for isomorphic copies) [7,8]. We may express the fact that Z ( H ) is finite by saying that H is a homomorphic image of “most” graphs G satisfying chr(G) I chr(H), or by saying that the properties H g G and chr(G) 5 chr(H) of a graph G are “almost equivalent” (cf. opening paragraph).

We shall use the symbol K(al, a2 , . . . , a,) to denote a complete n-partite graph with parts of cardinalities 1 5 al I a2 I ... I a,. Note that K ( a ) is the set of a isolated vertices.

We are now in a position to state our main result:

THEOREM. Let H be any graph. Then Z(H) is finite if and only if H is a complete multipartite graph, K(al, a2 , . . . , a,,), with

n = 1 and al any cardinal,

n = 2 and al I a2 < 00,

n = 3 and al = a2 = 1 I a3 < 00,

4 ~n < 00 and al = a2 = ... = a, = 1

or

or

or or H is a complete bipartite graph with an edge deleted, K- (a, b), with 1 < a < b < 00.

272 Annals New York Academy of Sciences

In order to establish this result we shall begin by reviewing some facts about the conjunction of graphs ([ 11, p. 25). The conjunction of two graphs G and H, denoted by G A H, is the graph with vertex-set V ( G ) x V ( H ) , in which the vertices (u, w ) and (v’ , w ‘ ) are adjacent if and only if the vertices v, v’ are adjacent in G and the vertices w, w‘ are adjacent in H . The projections p: G A H -+ G, q : G AH -+ H are mappings defined by p(u, w ) = u, q(v , w ) = w ; obviously, both p and q are homomorphisms. (Moreover, if neither G nor H is edgeless, then all mappings p, p’ , q, q x are surjective, so that G g G A H and H A G AH.) Consequently, the chromatic number of G A H does not exceed the smaller of the numbers chr(G), chr(H). (The question as to whether chr(G A H) actually equals min{chr(G), chr(H)} remains an open problem, [3, 6, 121.) If g: X + G and h: X + H are homomorphisms, then 4: X - G A H defined by 4(x) = (g(x), h ( x ) ) is also a homomorphism and p 0 4 = g. q 0 4 = h. (Furthermore, 4 is the unique homomorphism X -+ G AH with these properties [6, 121; for this reason the conjunction is often called the categorical product, or the product.) Note that the conjunction of irreducible graphs is irreducible. It is observed in [12, 141 that if both G and H are connected, and if chr(G) 2 3, then G A H is connected. To illustrate how these facts can be used, note that it follows that each graph K, A K , , t > 2, is a finite irreducible connected graph of chromatic number 2. Therefore, if chr(H) 2 2, then K, A K , E Z ( H ) , unless H 5 K, A K , , in which case H must also be finite and connected. Thus if Z ( H ) is finite, then H is a finite connected graph, or chr(H) = 1.

The case chr(H) = 1 is the easiest to dispose of:

PROPOSITION 1. Let chr(H) = 1. Then Z ( H ) is finite.

Proof: In fact, Z ( H ) C_ {K,}, as K, is the only irreducible graph of chromatic number 1. 0

Note that each graph H with chr(H) = 1 is isomorphic to some K(a) , where a may be infinite. We assume from now on, unless otherwise stated, that H is a finite, connected graph.

BIPARTITE GRAPHS

In this section we restrict our attention to finite connected graphs H satisfying chr(H) = 2. As noted above, Z ( H ) will be infinite unless H is a quotient of K t A K , , for all sufficiently large t > 2. We shall show that a bipartite quotient of any such K , A K 2 is a complete bipartite graph with a (possibly empty) matching removed. For t > 2, the graph X = K, A K , is connected, and hence uniquely 2- colorable. In fact, denoting the vertices of K, by 0, 1, . . . , t - 1 and those of K 2 by 0, 1, it is easy to see that c(i, j ) = j , i = 0, I, ..., t - 1, j = 0, 1, is the unique 2- coloring of X. Let X, denote the set of vertices colored j by c ( j = 0, 1). If C is a congruence on X with classes C,, C2, . , ., C , such that H = X/C is bipartite, then any 2-coloring of H induces a 2-coloring of X in which all vertices of each Ci obtain the same color. As X is uniquely 2-colored by c, this implies that each class Ci is a subset of Xo or of XI. Let C, E Xo and C, E X,; if Ci is not adjacent to Cj, then necessarily Ci = {(x,O)} and C j = {(x, 1)) for some x = 0, 1, . . . , t - 1. Hence H = X/C is isomorphic to some K(a, 6) with a (possibly empty)’set of disjoint edges deleted. Therefore we have:

(1) I f Z ( H ) is finite, then H z K ( a , 6 ) - M , where M is a matching.


This result can be improved to yield:

( 2 ) If Z ( H ) is finite, then H z K(a, b ) ur H 2 K - (a, b).

Let H r K(a, b ) - M and assume IMI 2 2; let {ul , u 2 , ..., uo}, { w l , w 2 , ..., wb} be the bipartition of H; let M = { [ u i , w , ] : i = 1, 2, ..., k}, k 2 2. For t 2 1, consider the graph A, as defined in FIGURE 2.

FIGURE 2. The graphs A , .

In order to show that each A, belongs to Z(H) , we first show that any bipartite quotient of A, is actually an induced subgraph of A,. Let C be a congruence on A,, with classes Cl, C2, . . . , C, such that AJC is bipartite. Then, as in the preceding proof, each Ci is a subset of (xlr x2, ..., x,} or of (yl, y,, ..., y,}. Let r(Ci) be the x j with the lowest subscript j such that x j E Ci or the y j with the highest subscript j such that y j E Ci . It is now evident that the subgraph of A, induced by the vertices r(Cl) , r(C2), . . . , r(C,) is isomorphic to the quotient A,/C. Now, in order to show that H is not isomorphic to any induced subgraph of A,, assume the contrary. Let u l , u2 correspond to x i , x j and w, , w2 to y , , y,. (Since H is assumed to be connected in this section, it is uniquely 2-colorable, and it cannot happen that ul corresponds to some xi and u2 to some y j . ) As xi is adjacent to y , but not to y , , we have r c i I s; as xi is adjacent to y, but not to y,, we have s < j I r. This contradiction proves that H 4 A,. Clearly, each A, is an irreducible bipartite graph, and hence Z ( H ) is infinite.

Note that the case H z K ( 1 , b ) does not belong to this section, as H is disconnected; according to the remarks preceding this section Z(H) is infinite provided b L 2.

For the remainder of this section, G will denote a graph of chromatic number 2, which admits some K , as a quotient. Let g : G -+ K, be a homomorphism under which K , d G, and let h: G -, K 2 be a 2-coloring of H (i.e., g, g’ , h, h’ are all surjective mappin s). Let 4: G -+ K, A K 2 be the homomorphism defined by 4 ( u ) = (g(u), h(u)), and let E be the homomorphic image of G under 4. In what follows, we shall assume that the vertices of K 2 are denoted by 0, 1, and those of K, by u, with x from some n-element set of subscripts.

(3) Ifu, # u y , then [ (v , , O), (uu, 111 or [(u,, I ) , ( u y , O)] is an edge o f c . Since g # is a surjective mapping E ( G ) -+ E(K,) , there exists an edge [w, w’] in G

such that g(w) = u x , g(w’) = u y . Since g = p 0 4, where p is the projection K, A K 2 + K,, 4 takes the vertices w, w’ to (u,, 0), ( t r y , 1) or to (D,, I), ( u y , O), respectively.

(4) I f K f o b s G, then K(a, b ) s G.


Continuing with the above notation, we let n = 2ab, and denote the vertices of KZd by u r n , , , , , for 01 = 1, 2, .. ., a, fl = 1, 2, ..., b, i = 0, 1. Let

C, = {(u,, ,, i , 0) E V ( C ) : fl = I, 2, . . . , b, i = 0, 11, for a = 1,2, . . . , a, D , = {(u,, ,, , , 1) E V ( G ) : a = 1, 2, . . . , a, i = 0, l}, for B = 1, 2, . . . , b,

and let C be the congruence on G with the classes Cl, C2, ..., C,, D1, D,, ..., Db. In G/C we have each Ca adjacent to each D,, because according to (3), [(ua. f i .o ,O) , (on, f i .1~0)l or [(urn, B, I , 01, (ua, p. Q, 113 is an edge of G. Obviously, no two vertices C,, C,. or D, , D,. are adjacent in G/C. Hence e/C z K(a, 6 ) g G_a G.

(5) I f n > 4k, thenfor some u, in K , , deg ( (u, , 0 ) ) 2 k and deg((u,, 1)) 2 k in G. Otherwise, let deg((u,, 0)) < k, for x E I, and deg((u,, 1)) < k for x E J, I n J = 4,

and II u J I = n. Let I I I = n‘ 2 42, and let e denote the number of edges between {(a,, 0): x E I } and ((0,. 1): x E I). By (3), deg((u,, 1)) 2 n - 1 - k, for each x E I. However,

n’ . ( k - 1) L e L n’ . (n - 1 - k - (n - n’)) = n’ . (n’ - 1 - k) ,

and so k 2 n’/2 2 n/4 > k. This contradiction establishes the claim (5) .

(6) Let 1 2 a I b and n = 2ab + a + 2b. 1 f K . s G , then K - ( a + 1, b + 1)s G.

Since n > 4b. some vertex v, of K, satisfies deg((u,, 0)) 2, b and deg((u,, 1)) L b 2 a (in c), by (5 ) . Let the b vertices (u , , I), y E Y, be adjacent to (u,, 0), and the a vertices ( u z , 0), z E Z , adjacent to (u, , 1) in G. Since I Y I = b and IZI = a, we have I Y + Z I I a + b. Hence there are at least n - 1 - a - b 2 2ab vertices u,, with s $ {x} u Y u Z, in K , . Consider a set of subscripts S satisfying IS1 = 2ab, S n ({x} u Y u Z) = 4, and the subgraph G* of G induced by the vertices ( u s , i), s E S , i = 0, 1. Then chr(G*) = 2 and K z d d G* under the restriction of the proiec- .tion p: K, A K , + K , to G*. According to (4), there is a congruence C* on G* such that G*/C* z K(a , b). Let Xi = {(u, i ) : u E K,}, i = 0, 1; Xo. X1 form the unique bipartition of K, A K , . It follows from the proof of (4) that the classes Cl*, C,*, . . . , C,* of C*, corresponding to the first part of the bipartition of K(a, b), may be chosen to be subsets of Xo, and the classes D1*, D2*, ..., Db*, corresponding to the second part, subset of XI. We shall define a congruence c on G: We extend each C,* by a different vertex (vz, 0), z E Z . (This is possible, as IZI = a.) We extend each D,* by a different vertex (u,, I), y E Y. (Recall that 1 Y I = b.) We create two new classes, { ( u , , 0)) and { (u , , 1)). Finally, we place all remaining vertices ( u , , 0) in an arbitrary existing class containing some ( u s , 0), s E S, and all remaining vertices ( u , , 1) in an arbitrary existing class containing some (u , , I), s E S. It is now easy to observe that the resulting congruence on G yields G/c z K - ( n + 1, 6 + 1). Hence K - ( a + 1, b + 1)s c s G. (We note for future reference that each class of c is a subset of Xo or of XI.)

PROPOSITION 2. Let chr(H) = 2. Then Z ( H ) is finite if and only if H 2 K(a, b ) or H z K - ( a + 1, b + 1) for some 1 5 a 5 6.

Proof: The necessity follows from (l), (2), and the remark following (2). For the sufficiency, note that (4) implies Z ( K ( a , b)) G Z(K,,) , (6) implies Z ( K - ( a + 1, b + 1)) E Z(Kz&+o+Zb). and that each Z ( K , ) IS finite by [8]. 0

Hell & Miller: Graphs with Forbidden Homomorphic Images

OTHER GRAPHS

275

Let n = chr(H) 2 3. Generalizing from the bipartite case, we observe that Z ( H ) will be infinite unless H is a quotient of K, A K,, for all sufficiently large t > n. Any such K , A K , contains K, as a subgraph, and hence so does any quotient of K, A K,. Moreover, it is easy to see that each K, A K,, with t > n, is uniquely n- colorable; if H is a quotient of K, A K, with chr(H) = n, then H is also uniquely n-colorable, since distinct colorings of H induce distinct colorings of K , A K , . We have:

( 7 ) If Z ( H ) isfinite, then H is uniquely n-colorable, and contains K , .

We note that by an argument analogous with the proof of (l), one can deduce that if Z ( H ) is finite, then H z K ( a , , a 2 , ..., a,) - L, where L is the set of edges of a vertex-disjoint union of complete subgraphs.

Let K,(i) denote the graph obtained from K, by adjoining a new vertex adjacent to exactly i vertices of K , .

(8) If i I n - 2, then Z ( K , ( i ) ) is infinite.

Each such K,(i) admits 2 distinct n-colorings, and so (8) follows from (7).

(9) Ifn 2 4, then Z(K,(n - 1)) is infinite.

Consider the graphs B,,', n 2 4, t 2 1, defined as in FIGURE 3.

1 n

Each xi is adjacent to 1, 2, ..., [n/2 j; each y , is adjacent to [n /2 j + 1, ..., n ; x i , y i a r e a d j a c e n t o i # j ; 1, 2, . . ., n, form a complete graph.

FIGURE 3. The graphs B t .

Each B,,' E Z(K,(n - 1)): It is a simple exercise to show that each B,,' is irreducible and that chr(B,,') = n. To show that K,(n - 1)$ B,' we assume the contrary and let C be a congruence on B,,' such that B i / C K,(n - 1). Let C1, C2, . .., C, be the congruence classes corresponding to the vertices of K , and Co the class corresponding to the new vertex (of K,(n - 1)). We may assume that each vertex i , i = 1,2, . . . , n, belongs to the congruence class Ci . There are only three possible locations for Co: either Co = { x i , yi}, for some i = 1, 2, ..., t , or Co G {x,, x 2 , ..., x l } , or else Co s { y , , y 2 , ..., y,}. In the first case B,,'/C g K , + , , in the second case B,,'/C z K,([n/2J), and in the third case B,,'/Cz K,([n/21). This is a contradiction.

(10) Let chr(H) = chr(H'). If H g H' and Z ( H ) is infinite, then so is Z(H').

Indeed,Z(H) = {G E 9: Girreducible,chr(G) 5 chr(H), H irreducible, chr(G) I chr(H'), H'$ G}.

G} G Z(H' ) = {G E 9: G


(1 1) If n = chr(H) 2 4 and H $ K , , then Z(H) is infinite. Let Cl, C 2 , . . . , C, be an n-coloring of H; since H $ K,, some color-class, say

C1, is not a one-element set. Let C , = C1' u C1" be a nontrivial partition of C1. Let C be the congruence with classes C,', C1", Cz, ..., C , . In the quotient H / C , any two classes Ci, C j , with i > j 2 2, are adjacent (because chr(H) = n), and each Ci, i 2 2, is adjacent to C1' or to C1" (for the same reason), If H contains no K,, then Z(H) is infinite, by (7); otherwise, H/C z K,(i) for some i = 1, 2, . . . , n - 1 and Z(H) is infinite, by (8), (9), and (10).

PROPOSITION 3. Let chr(H) 2 4. Then Z(H)is b i t e if and only if H E K ( 1, 1, . . . , 1).

Proof: Let n = chr(H). Then Z(H) is infinite unless H z K, 2 K(1, 1, . .., l),

(12) Let chr(H) = 3, and suppose thatfor some 3-coloring of H there are at least

by (1 I), and Z ( K , ) is finite, by [8]. 0

two edges between any two color classes. Then Z(H) is infinite.

1 2

Vertices 1 and 2 are adjacent. Each xi is adjacent to 1, 2, and yi. t

. . .

FIGURE 4. The graphs C, .

Consider the graphs C, , t 2 1, as defined in FIGURE 4. Each C, is irreducible and satisfies chr(C,) = 3. To prove that C, E Z(H), it remains to show that H $ C,. If H were a quotient of C,, then a 3-coloring of H with at least two edges between any two color classes would induce a 3-coloring of C, with the same property. However, it is easy to see that C, does not admit such a 3-coloring.

(13) I f c h r ( H ) = 3, and Z(H) isfinite, then H 2 K(1, 1, b). According to (7), H contains three mutually adjacent vertices, say x, y , z. Let c be

any 3-coloring of H, and let X, Y, Z be its color classes. (We may assume that x E X, y E Y, and z E Z.) By (12), we may assume that, say, [x, y ] is the only edge between X and Y. Note that H is connected (since Z(H) is finite). If X contains a vertex x' # x, then the congruence C with classes {x'}, X - {x'}, Y, Z yields H/C z K 3 ( 1) and Z(H) is infinite, by (8) and (lo), contrary to our assumption.

A similar contradiction results if Y contains a vertex y' # y , or if Z contains a vertex z' # z adjacent to either x or y . In the only remaining case, Z contains only vertices adjacent to both x and y . and X = (XI, Y = { y ) . Hence H r K(1, 1, b) , for some b 2 1.

(14) Z(K( 1, a, b ) ) contains only finitely many bipartite graphs.

This follows from PROPOSITION 2 and the observation that K(1, a, b ) s

(15) Let chr(G) = 3 and b 2 1. If K 7 b - z ~ G, then K(1, 1, b ) s ~ G .

K - ( a + 1, b + 1).


Let n = l b - 2. Let g: G -+ K , and H: G -+ K 3 be homomorphisms yielding K, and K , as quotients of G (i.e., all mappings g, g’, h, h X are surjective). Let 4: G K, A K , be the homomorphism defined by 4 ( u ) = (g(u), h(u)) and let e be the homomorphic image of G under 4. Thus G is a subgraph of K, A K , ; we shall denote the vertices of K3 by 0, 1, 2, and those of K, by u l , t i z , ..., u n . Let X j = { ( u i , j ) : i = 1 , 2 ,..., n } , j = O , 1,2.

If it is possible to partition some X , into b subsets X,”, a = 1, 2, . . ., b, so that each Xp is adjacent (in c) to both sets X i with i # j , then the congruence with classes X p (a = 1, 2, . . ., b), X i (i # j ) , yields K(1, 1, b) as the quotient.

Otherwise, for each X , , j = 0, 1, 2, there exists an Xf(,), with f(j) # j , such that fewer than b vertices of X j are adjacent (in e) to X,,,?; denoting

I j = { i : 1 I i 2 n, ( u i , j ) is adjacent to X f c j ) } ,

we have JIjJ < b,forj=O, 1,2. Therefore the set N = {l, 2, ..., n} - (Io u II u I , ) has at least 4b + 1 elements.

Iff is a permutation, then it is a 3-cycle, andf(j) # f- ’ ( j ) , for each j = 0, 1, 2. Since 4b + 1 2 5, we may assume that there are distinct elements x and y in N. By the definition off, each vertex (D,, j ) is not adjacent to ( u y , f ( j ) ) and each vertex (uyr j ) is not adjacent to (u,,f(j)), j = 0, 1, 2.

Hence, each ( u , , j ) is not adjacent to ( u I , f ( j ) ) , ( u Y , f - ’ ( j ) ) , and ( u y , j ) (because G is a subgraph of K, A K , ) . But {f(j), f- ( j) , j } = (0, 1, 2); therefore no (u,, j ) is adjacent to any ( u y , i). This contradicts the facts that 9’: E(G) -+ E(K,) is surjective (cf. the proof of (3) for details).

Iffis not a permutation, then, without loss of generality,f(l) = O,f(2) = 0, and f(0) = 1. Let N* be a subset of N with exactly 46 + 1 elements and M*, the subgraph of G induced by the vertices { ( u i , j ) : i E N*, j = 1, 2). Consider any two distinct elements x, y of N. Since [ (u , , 0), ( u , l)], [(u,, l), ( u , O)] are not edges of G, by definition of Io, and [ (u, , 0), ( u y , 2)i [(u,, 2) ( u y , 0)f are not edges of c, by the definition of 12, it follows from the surjectivity of g x that [ (u, , I), ( u y , 2)] or [ (u, , 2), (uy,, l)] is an edge of M*. Consequently, K . + b + l g M * under the restriction of the projection p: K , A K , + K , to M*, and according to (6), with a = 1, K-(2, b + 1 ) g M*. Furthermore, it follows from the proof of (6) that we may choose a congruence C* on M* yielding M*/C* z K-(2, b + 1) in such a way that the two classes Co, C1, corresponding to the first part of the bipartition of K ( 2 , b + l), are subsets of X1, and the b + 1 classes D o , D1 , ..., D b , corresponding to the second part, subsets of X,. (We assume that there is no edge between Co and Do corresponding to the deleted edge of K-(2, b + l).) Now we define a congruence c on G to have the classes

Co U Do U Xo, Xi - Co, D1, D2, ..., D b - 1 , X 2 - (Do U D1 U .‘. U & - I )

It is a simple task to verify that each class of G/c 2 K(1, 1, b), FIGURE 5.

is an independent set and that

PROPOSITION 4. Let chr(H) = 3. Then Z ( H ) is finite if and only if H 2 K( 1, 1, b) for some b 2 1.

Proof: The necessity follows from (13), the sufficiency from (14) and (15) in conjunction with [8] .

Our main theorem follows from the preceding results.


(Vx,O) (v ,O) Y I - - -

M

FIGURE 5. An illustration of the last part of the proof of (15).

REMARKS

According to our theorem, Z ( K , ) is finite for each finite complete graph K,. This result, first proved in [8], implies that there are only finitely many irreducible graphs with any given finite achromatic number. To see this, note that Z ( K , ) = {G: G irreducible, chr(G) 5 n, and K,$ C } equals {G: G irreducible, achr(G) < n}, because, by the Interpolation Theorem of [l], p. 144, [2], (it is easy to extend the theorem to infinite graphs as well), the statement, chr(G) n and K,$ G, is equivalent to the statement, achr(G) < n. It follows that the number of vertices of an irreducible graph with a given (finite) achromatic number is bounded. (This fact can also be deduced from Lemmas 4.2-4.5 of [9].) Hence we are led to ask for the smallest achromatic number f ( n ) among all irreducible graphs with n vertices. A rough analysis of the proof in [8] shows thatf(n) > ,,h* A recent result of Mate [ll], establishes that in factf(n) 2 (1/2 - o(1))log n/log log n, while examples suggested by P. Erdos show that f ( n ) I log n/log 2 + 2 [7, 111. It would be interesting to obtain better bounds for f .

Let us denote the statement “there is a homomorphism G -. H ” by H I G, and its negation by H $ G. In analogy to our analysis above, one could study the equivalence of (H I G) and (chr(G) 5 chr(H)). In fact, let

Z ( H ) = {G: G irreducible, chr(G) 5 chr(H), H $ G}.

The situation here is much less interesting, and we find that the following statements are equivalent:

(i) Z ‘ ( H ) is finite; (ii) chr(H) = cl(H), where cl(H), the clique number of H, is the cardinality of a

(iii) Z’(H) = 4. maximum clique in H ;

In other words, given a graph H, the equivalence (H I G)o(chr(G) I chr(H)) either holds for all graphs G (when chr(H) = cl(H)), or fails for infinitely many irreducible graphs G (when chr(H) > cl(H) = n or when H is infinite and cl(H) is undefined). In fact, when cl(H) = n < chr(H), then K , , with a path of an arbitrary length emanating from one vertex belongs to Z ( H ) ; when H contains complete subgraphs of cardinalities a, but no complete subgraph of cardinality a = lim a,, then K , with a parth of an arbitrary length emanating from one vertex belongs to Z’(H) .


When Z(H) is finite, an opportunity offers itself to give a simple description of all graphs G for which H is a forbidden homomorphic image. In the remainder of the paper, we shall do this for all such graphs H, with fewer than five vertices.

Let Z ( H ) be finite. The class of graphs G for which H is a forbidden homomorphic image consists of F ( H ) and, of course, of all graphs G with chr(G) > chr(H). The latter will be omitted from the descriptions below. To describe F(H), one may begin by describing all the irreducible graphs of F ( H ) , i.e., Z(H) , and then simply label the vertices of such graphs by the admissible ranges of the size of the congruence classes (of the reducing congruence). For example, 1 -0 1 - 00

describes the family of graphs for which the reducing congruence has exactly two classes. The classes are adjacent, one class is a singleton, and the other is of any finite or infinite cardinality. Clearly, these are the graphs obtained from the depicted graph by replacing each vertex with a set of vertices whose cardinality is within the indicated bounds and joining two vertices if and only if the vertices they replace are adjacent. In our present example, we obtain all (finite and infinite) stars. (The ranges are understood to be inclusive, e.g., 0-2 stands for (0, 1, 2}.)

H F (H)

A

A 17

1 1 1 1 1-2 - 1 - 2 . : = *

1-1

1-1

1-1

1-1

1 1 1 1-- 1 1 1 1 1 1--- 1--

lD---o 1-2 1--1--

1- 1-2 1-1 1-1

1-1 1-1

1-1 1-1-2

1-1

1-1 1-1 1

1-1

FIGURE 6. Graphs with forbidden homomorphic image H.


In FIGURE 6, certain graphs H with finite Z(H) are omitted. They include the graphs H of chromatic number 1. Clearly F ( H ) = { K ( a ) : 1 5 a I 1 V(H)I - l} for such graphs. Furthermore, F ( K , ) = (K(a) : 1 s a a}, and so K , is also omitted from FIGURE 6. Lastly, we omitted K , , because F(K,) is described in FIGURE 2 of [8]. Note that for chr(H) 2 2, we have G u K , E F ( H ) if and only if G E F ( H ) ; hence we agree to depict only those members of F ( H ) that do not have isolated vertices.

To illustrate the methods used in deriving the above results, we shall derive F(C4). We shall use the following concepts: The diameter of a connected graph G is defined by diam(G) = sup{d(g, g’) : g, g’ E V(G)} , where d(g, g’) is the distance from g to g’ in G. If G is an induced subgraph of G, a retraction r: G + G is a homomorphism such that r(g‘) = g’, for each g’ E V ( G ) . If there is a retraction r: G -+ G‘, then G is called a retract of G; clearly, each retract of G is a homomorphic image of G. It follows from [4, 5, 131 that in a bipartite graph G any subgraph of the following type is a retract of G: (A) a cycle of the shortest length, (B) a path from g to g’ of the shortest length, (C) a complete bipartite subgraph.

Let G be a connecfed graph in F(C,). If diam(G) 2 2, then G is a complete bipartite graph K(a, b); if 2 I a I b, then C , is a subgraph of G, and hence C 4 g G, by (A) or (C). Thus G is a star, and clearly each star belongs to F(C4). If diam(G) 2 4, then P o , the path of length 4, is a retract of G by (B), and C 4 a P 4 9 G. Hence it remains to study graphs G with diam(G) = 3. If G contains a cycle, then according to (B) and in view of C n q Cn+,, we have C 4 s G. It is easy to see that each tree of diameter 3 has as its type the path of length 3, and that all such trees are in F(C,).

If G is a disconnected graph in F(C,), then each component is a connected graph in F(C,). Now it is easy to complete the description of F(C,).

We remark that Theorem 4.2.iii of [13] is used in the analysis of F(K(1 , 1, 2)). Note that F ( K 3 ) = F(P3) . This is a very special case of a general result due to KO [lo].

REFERENCES

1. HARARY, F. 1969. Graph Theory. Addison-Wesley. 2. HARARY, F., S. HEDETNIEMI & G. PRINS. 1967. An interpolation theorem for graphical

homomorphism. Port. Math. 26: 454-462. 3. HEDETNIEMI, S. 1966. Homomorphisms of graphs and automata. Univ. Mich. Tech. Rep.

4. HELL, P. 1972. Retractions de graphes. University of Montreal. Doctoral dissertation. 5. HELL, P. 1974. Absolute retracts in graphs. In Graphs and Combinatorics. R. Bari &

6. HELL, P. 1979. An introduction to the category of graphs. In press. Ann. N.Y. Acad. Sci. 7. HELL, P. & D. J. MILLER. 1976.011 forbidden quotients and the achromatic number. Proc. 5th

British Combinatorial Conf. Congressus Numerantium XV. Utilitas Math. pp. 283-292. 8. HELL, P. & D. J. MILLER. 1976. Graphs with given achromatic number. Discrete Math.

9. HOFFMAN, A. 1972. Eigenvalues and partitionings of the edges of a graph. Linear

03105-44-T.

F. Harary, Eds., Springer Lect. Notes Math. 406: 291-302.

16: 195-207.

Algebra Appl. 5: 137-146. 10. KO, C. S. Personal communication. 11. MA* A. A lower estimate for the achromatic number of irreducible graphs. To appear. 12. MILLER, D. J. 1968. The categorical product of graphs, Can. J. Math. 20: 1511-1521. 13. SABIDUSSI, G. 1973. Subdirect representations of graphs. In Infinite and Finite Sets.

14. WEICHSEL, P. M. 1962. The Kronecker product of graphs. Proc. Am. Math. SOC. 13: 47-52. Keszthely Colloq. Math. Soc. Janos Bolyai. 10: 1199-1226.

graphs with forbidden homomorphic images

Documents