A generalization of Turán's theorem to directed graphs

Di%.ete Mathematics 32 (1980) 167-189 @ Earth-Holland Publishing Company zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A GENERALHZATION OF TURh’S TO DIRECTED GRAPHS THEOREM Stephen B. MAURER* Department of Mathematics, Swarthmore College, Swarihmore, PA 19081, USA lssie RABINOVITCH DeparIment of Mathematics, Concordia University, h?ontred, Quebec, Canada H3G lM8 William T. TROTTER, Jr. Department of Mathematics and Statistics, Univ. of Sleuth Carolina, Columbia, SC29208, Received 8 January USA 1980 We consider An extremal problem for directed graphs which is closely related to Tutin’s theorem giving the maximum number of edges in a gr;lph on n vertices which does not contain a complete subgraph on m vertices. For an ;ntc&r n 22, let T,, denote the transitive tournament with vertex set X,, = {1,2,3, . . . , n) and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON edge set {(i. j): 1 s i C j s n]. A subgraph H of T,, is said to be m-locally unipathic when the restriction of H to each m element subset of X,, consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T,, is (;g)(m - l)‘+q(m - 1)r + Ur”] where n = q(m - 1) + r and [$<m - I)] s rc’ @rn - 1)1. As is the case with T&n’s theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike T:r&n’s theorem, the part sizes will not be uniform The proof of our principal theorem rests 011a combiaatorial theory originally developed to inves:dgate the rank of partial’iy ordered sets. For integers, n, k with zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB n se k a 2, let g(n, k) be the maximum number of edges in a graph G on n vertices which does not contain a complete subgraph on k vertices. Then let n = (k - 1)q + r where 0 6 r C k - 1 and consider the complete multipartite graph G(n, k) having k - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH 1- r parts of size q and r parts of size q f 1. Clearly, G(n, k) has n vertices but does not have a complete subgraph on k vertices. The following well known theorem of P’. Tur6n [9] tell us that the lower bound on g(n, k) provided by the graph G(n, k) is best possible. It also Sells us that G(n, k) is the unique extremal graph. Theorem 1 (Turhn). For integers m,k with n > k 2~2 the maximurn number g(n, k) of edges in a graph on n vertices which does not contain a complete subgraph on k *Work supported by United States National Science Foundntion Grant MCS 76-19670 while the author was in the Mathematics Department at Princeton University. 167 S.B. M aurer, 1. Rabinovitch, W.T.Trotter, Jr. vertices is given by : (‘k-1-r gIn,k)= “r 2 wheren=(k-l)q+rand )q2+0; Oar<k (q + l)*+ (k - 1 - 1. Furthermore, - r)rq(q + 1) if G is a graph on n vertices jvhich does not contain a complete subgraph on k vertices, then G has g(n, k) edges if and only if G = G(n, k). In this paper, we will consider a similar combinatorial problem involving the maximum number of edges in a directed graph which satisfies a particular property. As in Tur6n’s theorem, the extremal graph(s) will complete multipartite graphs, although the part sizes will not all ke uniform. For an integer n 2 2, let T, denote the Vansitive tournament with vertex set X,=(1,2, ,..* n) and edge set ((i, j): 1 s i C,i C n}. A subgraph H of T,, is said to be unipathic if for each pair x, y of distinct vertices, H contains at most one directed path from x to y. Now consider the following elementary extremal problem: What is the maximum number u(n) of edges in a unipathic subgraph of T,? It is easy to see that this problem is equivalent to a special case of Turhn’s theorem. Thearen 2. Far each n a 2, the maximum number u(n) of edges in a unipathic subgraph of I’, is given by the formula: u(n) = [in*J. Furthermore, if H is a unipathic subgraph of T,, having u(n) edges, then the underlying undirected graph derermined by H is the compjeete bipartite graph K( [$I, [$nl); M oreover, zyxwvutsrqponmlkjih if n >4, the verbces in each of the two parts of H occur consecutively in {1,2,3, . . . , n}. Pro& Let H be B unipathic subgraph of T,, and let G be the underlying undirected graph detixmined by H. Since H is unipathic, G is triangle-free, i.e., G does not contain KS. Thus H and G have at most g(n, 3) = [$n*J edges. On the other halcld, let t = 1.4n j (or t = [$nl ) and consider the subgraph H of T, containing the kedges {(i, j) : 1 L i s t, t + 1 ~j Q n}. Clearly , H is unipathic and contains En”J edges, and thts u(n) = [$z’J. Finally, suppose that n 24 and let H be a unipathic subgraph or’ T, containing Un’J ejges. It foll ows from Tur&n”s theorem that the underlqi mg undirected graph C determined by H is the complete bipartite graph K( [$z], [$I). Then let A and B denote the subsets of {1,2,3, . . . , n} which form the vertex sets of the two parts of G. If n 24 and either .:‘? or B does not occur consecutively in n}, then there exists integers a,, a2E A, bl, b2E B for which one of {1.2.3...., the following statements holds: a, C 6: c a2 < b2, al c bl SCb2 C a*, 6, < a1 < a2 < bz, or bl <a, -=z b2<a2. HIIeach of the four cases, H would fail to be unipathic even though it is triangle-free. -We note that although u(3) = !5*/4] = 2, there are three extremal graphs A generalization of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE T ush’s theorem to directed graphs ii? corresponding to A = {l}, A = {1,2} and A = (1,3} respectively. {(1,2), U,3)1, {(1,3), (2,3)), and {(L 2), (2,3)). These graphs are q Now let n and m be integers with IZa m 3 2. A subgraph H of zyxwvutsrqponmlkjihgfedcbaZY T, is said to be m-locally unipathic when the restriction of H to each subset of V, containing m consecutive vertices is unipathic. On the other hand, H is said to be m-locally kangle-free when the restriction of H to ,ach subset of X, containing m consecutive vertices is triangle-free. Then let u(n, m) be the maximum number of edges in an m-locally unipathic subgraph of T,, and A (n, m) the maximum number of edges in an m-locally triangle-free subgraph of T,,. We have already observed that u(n, n) = A(n, n) = g(n. 3) = [&z’J for every n 2 2. Furthermore, it is easy to see that u(n, 2) - A (n, 2) = (;) for every n 3 2 and that A(n, m)a u(n, m) for every ii 2 m 22. In view of Theorem 2, it is reasonable to conjecture that the extremal graphs for u(n, m) are complete multipartite graphs for all n k m 3 2 with the vertices in each part occurring consecutively in {1,2,3, . . . , n} (except for the case (n, m) = (3,3)). Analysis of the properties of such graphs suggests the following with n=q(m-l)+r>2, we scheme. For arbitrary integers m ~=2,qbO,rbO construct a complete multipartite subgraph H(m: q. r) of T,. We begin by setting V,, = CL 2,3, . . . , l&J>, Vq+l = {n - [$rl + 1, n - [$I + 2, n - [&I+ 3, . . . , n}, and Vi={(i-l)(m-l)+[$r]+j:lGjCm-l} for i=l,2,...,q. Finally, we define H(m, q, r) as the complete multipartite graph having q + 2 parts with edge set ((j,, j2): There exist i,. i2 with j, E Vi,, j2~ Vi2 V,, v,, v2, * * * , vq+, and 0 s i, < i2 --=q + 1). Note that H(m, q, r) is a bipartite graph when q = 0. Also note that V, contains the first L$j vertices of (1,2,3, . . . , n) and V4+, contains the last [irl vertices of {1,2,3, . . . , n}. We then denote by &m, q, r) the complete multipartite graph obtained by reversing the roles of V, and Vq+lr i.e.. in &m, q, r), V,, contains the first [$rl vertices of {l, 2,3, . . . , n} and V4+1 contains the last [$J. Note that K(m, q, r) and fi(m, q, r) have the same number of edges. In fact, H(m, q, r) = fi(m, q, r) when r is even. Fo:: convenience, we let h(m, q, r) denote the number of e,>ges in H(m, q, r). Note that h(m, q, r) = ($(m - 1)2+q(m - 1)r + l$r’]. easy to see that H(m, q, r) and fi(m, q, r) are m-locally unipathic. Furthermore, it is straightforward to verify that for fixed values ol n a nd m w it h 30 and ra0 so that n = q(m -- l)+ r, then the maximum n>m>2,ifwechooseq v&e of h(m, q, r) is achieved when [$(m - 1>1~ r < [$<m - 1)1. When m 2 3, this maximum value is achieved by a unique triple (m, q, r) unless r = f(m - 1) in which It is S.B. Maurer, I. Rabitwuitch, W.T. Trotter,Jr. 170 case, we have h(m, q, r) = h(m, q- 1, r+m - 1). For example H&3,2) and H&2,6) each have h(5.3,2) = h(5,2,6) = 73 edges; and H(l1, 1,5) and H( 11, 0.15) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA each have 56 edges. When m = 2, we observe that the maximum value of h(m, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA q, I) is (!!), and this maximum value is achieved if and only if r is 0, 1, or 2. We also observe that H(2, n, 0) = H(2, n - 1, 1) = H(2, n -2,2) = T, for every n a 2. These observations are summarized in the following result. Tkowm 3. Let n am 32. Then the maximum number u(n, m) of edges in an m-locally unipathic subgraph of ‘I’,, satisfies the inequl?lity: u(n, m) 2 h(m, q, r) = m - 1)2+eI(m - l)r+ lar*J, where n=q(m-l)+r In th: remaining and [f(m-l)lsr<[$(m-1)l. sections of this paper, we will show that the inequality in Theorem 3 is best possible. We will also show that if (n, m)# (3,3), then the complete multipartite graphs H(m, q, I) and fi(m, q, r) are the only extremal graphs except when r = $(m - 1). When r = $(m - 1) and r is even, there are two extremal graphs: H(m, q, r) and H(m, q - 1, r + m - 1). When r = $(m - 1) and r is odd, there are four extremai graphs: H(m, q, r)_ Z%(m,q, r), H(m, q - 1, r + m - 1) and &m, q - 1, r+ m - 1). Sections 2 and 3 will be devoted to the theoretical preliminaries, and the proof of the principal theorem will be presented in Section 4. In Secticn 5, we will present a brief discussion of the concept of rank for partially ordered sets and the specific problem which motivated our investigation of m-locally unipathic subgraphs of T,. 2. ‘Be cfigrapb of nonforcing p&s for a partially ordered set In this paper. a partially ordered set @set) is a pair (X, P) where X is a finite set and P is an irreflexive transitive binary relation on X. The notations (x, y) E f, x > y in P, and y <x in P are usecl interchangeably. The notations x d y in P andy~xinPmeanx>yinPoIx~yandwewritex~yinPwhenx#y,xjt:y in P, and y$x in P We also let 1, -=(ix, y): x ly in PI. A poset (X, P) is called a totally ordered set (also a linearly orQ:“r:d set or chain) when Ip = 8. Throughout this paper, we adopt the: folicl*;;ng conventions concerning directed graphs. We denote an edge from ;t vertex x to L vertex x to a vertex y by (x, y) and we specify a digraph by its edge zet. It is then understood that the vertex set of a digraph, when not explicitly described, is the set of endpoints of the edges. We may therefore view a binary relation on a set X as a digraph. For example, when al, a2, . . . , 4+l are distinct, we say that the sequence {(a,, a,+i): 1 d i s t} is a dir+cred path of length t from cr, to a,,,. When a,, a2,. . . , a, are distinct and A generalization of T&n’s theorem to directed graphs 171 we say that the sequence {(a,, Qi+i): 1 G i G t} is a zyxwvutsrqponmlkjihgfedcbaZYXW directed cy cle of length f. A digraph H is said to be acyclic when it does not contain any directed cycles. A digraph H is said to be unipathic when it contains at most one directed path from x to y for every pair of vertices x, y, i.e., if Pi = ((4, q+,): 1 C i c t} and P2 = ((vi, vi+,) : 1 sj bs} are paths in EZ, u1 = u1 = x, and y+l = us+l = y, then s = t and (h,ui+i) = (ui, ui+i) for i = 1,2, . . . , t. To assist in distinguishing directed and undirected graphs, we will continue the notational convention adopted in Section 1. Specifically, we will use the letter G to denote an undirected graph and the letters H and N to denote directed graphs. We will then use “primes” or subscripts when we are discussing more than one such graph. W hets X is a set, we let 1x1 denote the number of elements in X, and when H is a digraph, we let IHI denote the number of edges in H. Now let (X, P) be a poset and let (x, y)~ ZV We say that (JC,y) is ;7 conforcing pair when P W{(x, y)} is a partial order on X, i.e., z >x in P implies z > y in P, and z < y in P implies z <x in P for every z E X. We then let NP be the digraph (binary relation) of all nonforcing pairs. To illustrate this definition, we provide zyxwvutsrqpon in Fig. 1 the Hasse diagram of a poset (X, P) and the digraph N,, associated with a1 = ~f+~, (X7 P). Note that in general, the digraph NP may contain directed cycles. In order to extract an acyclic subgraph of NP, we adopt the following convention for “break- zyxwvutsrqpo iing ties”. Let L be an arbitrary linear order on X. Then define the acy clic digraph of nonforcing pairs I$ by It is straightforward to veri:‘y that Nf is acyclic, so we may adopt the same convention used for Hasse diagrams in providing a diagram for G. i.e., we require that each edge (a, b) E N$ be represented by a nonhorizontal arc with the point corresponding to a having larger y-coordinate then the point corresponding to b. In order to avoid drawing arrows, it is then understood that the direction of an edge is from top to bottom on the page. For example, if L is defined for the poset drawn in Fig. 1 by a > b > c > d > e >f > g > 12in L, then we may represent N$ as shown in Fig. 2. a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA h C zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC e f 5i h (X,P) Fig. 1. S.B. Maurer, I. Rabinouitch, W.T. Trotter, l’r. 172 h Fig. 2. The notation N$ does not indicate the particular linear order L used in its detinition since it is easy to see that the subgraphs of IVp determined by different linear orders Lre isomorphic. A subgraph H c ?$ is said to be unipathic relative to P (we also say H is a LJ$ graph) when the fo’llowing condition is satisfied: For each pair x, y of distinct vertices. if H contains two nonidentical paths from x to y, then (x, y) 4 IV:. For example, the subgraph HG G shown in Fig. 3 is unipathic idative to P (but it k not unipathic). Note that H contains nonidentical paths from a to d, but (a. d)d N$. In fact tl >d in P. We next present some elementary but important lemmas which detail the interplay between the partial order P and the acyclic digraph of nonforcing pairs. The proofs are immediate consequeuces of the definitions and are therefore omitted. Lemma 4. PfIIv$=@ l.emma 5. .PU A$ is a partial order on X. and (LQ,,4,)~ P for some io, i, with Ltmma 6, if ((4, &+,): I siGt}sPUIV$, 1 G i. C i, %t+l, then u,>q+, in P. It follows from kmmas 4 and 5 that a subgraph H of LG is a L$ graph if and only if it satisfies the following condition: For each pair X, y of distinct vertices, if Ii contains two nonidentical paths from x to y, then x > y in P. ‘These lemmas b C e f am d (X,Pi Np Fig. 3. ii = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM N; A generalization of Turhn’s theorem to directed graphs 173 also allow us to ma1.e the following observation concerning graphs whiich are not U$ graphs. If H c N$ and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED H is not a U$ graph, then there 1exisi.s an edge (x, y) E iV$ for which H contains two nonidentical paths P, and F$ fyorn x to y. Although these two lhaths are nonidentical, they may have vertices in common other than x and y and may also have common edges. On the other hand, if we examine all edges (x, y) of N$ for which H contains two or more nonidentical paths from x to y, and then choose an edge (x, y) E G and nonidentical paths P, and P2 from x to y for which the sum jPlj+ lP,j of the lengths of P, amd P2 is as small as possible, then it is easy to see that PI and Pz have no vertices in common other than x and y. A Ug graph H is called a maximal U$ graph when there does not exist a U$ graph H’ whose edge set contains the edge set of H as a proper subset. A L$ graph H is called a maximum e graph when zyxwvutsrqponmlkjihgfedcbaZ nc:ilC$ graph contains more edges than H. Maximal and maximum @ graphs are important concepts in the theory of rank of partially ordered sets, and we refer the reader to [a- 61 for details. In particular, we note that (except for certain degenerate cases) the rank of a partially ordered set (X, P) equals the number of edges in a maximum Us graph. In Section 5, we will return to this concept and employ the solution of our extremal problem to compute the rank of a class of partially ordered sets. 3. Exchange theesrems for tr;” graphs In this section,, we develop two exchange theorems for Ug graphs. These theorems establish conditions under which it is possible to exchange edges between a G graph H and lV$- H so as to produce a new U$ graph. Theorem 7. Let (X, P) be 4 poser and {a,, a,, a,} a subset of .X for which {(a,, a,), (a,, a,), (a,, a,)}~ G. If H is u U$ graph and {(a,, a,)~, (a,, a,)}~ H, !hen aYe U$ and H”= (H - {(a,, a*))) U {(a,, a,)} H’ = W - {(a,, a,))) 1J{(a,, a,)} graphs. Proof. We show that H’ is a Ug graph. The argument for H” is dual. Suppose to the contrary that H’ is not a t$ graph. Then there exists an edge (x, y) E N$ for which H’ contains nonidentical paths P,={(u,, 4+l): lcict} and P2={(2)i, oitl): l~:jss} from x to y. Without loss of generality, we may assume that the edge (x, y)~ N$ and the paths PI and P2 have been chosen so that s + t is mjnimum. We may then assume that x and y are the only two point; belonging to both P, and P2_ Since H is a LJ$ graph, we may assume without EDSSof generality that (a,, u3) E P,, say (a,, a,) = (Q z+,+,). Then it follows that H contains the paths Pi = (P- {(a,, u3)})U{(u,, a,), (a,, as)} and P2 from x to y and we must therefore have Pi = P2, which is impossible. The contradiction completes the proof. q S.B. M awr, I. Rabinovitch, W .T. Trotter,Jr. 1 7 4 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA a* a a b b b C C c zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA d, d d e, e e f f7 fi H2 5 0 % 3 *4 Fig. 4. To illustrate the preceding theorem, consider the poset (X, P) shown in Fig. 3 and the sequence of L$ graphs shown in Fig. 4. Observe that H2+, is obtained from Hi for i = 1,2,3 by an exchange permitted by Theorem 7. Also note that H, is a maximal (but not maximum) U’$ graph, but that Zf7: is not maximal since Hz U {(b, d)] is also a G graph. Therefore, an exchange of edges permitted by Theorem 7 may destroy the property of being a maximal L$ graph. For brevity, we say that a U$ graph H does not admit a Type 1 exchange when ((a,, a,), (a,, a3)r H implies (a,, a,)# .@ (and therefore aI >a3 in P) for all ul, u2, USE X. Our next exchange theorem describes a somewhat more complicated exchange. Theorem 8. Let (X, P) be a poser, {al, u2, u3, a& E X, and A = {(a,, ui) : 1 <i c j ~4) s A$. Further, suppose that H is a G gruph for which {(a,, a,), (a,, a,)) c H, and then let G(u2) = {z E X :(z, a,) E M and (z, a,) E ZV$- &) und L,(u,) = If H dues not admit u Type 1 {w ~X:(u~, W )EH and (u2, W )EN I- H}. exchange, {k tbzn the graph H’= (H- ((z, u2): z E G(u,)}- {:(a,, us): z E G(u2)) U {(u,, w ): w E L(u,)} is a U$ graph. w ): w E L(u,)}) u Froof. Suppose to the contrary that H’ is not a U$ graph. and choose an edge ix, y)~ h$r for which H’ contains nonidentical paths P1 = {(,h, h+i) : 1 =Zi d t} and P2 = {(ui, Vi.+,: 1 ,Cjgs} from x to y. As in Theorem 7, we assume that the edge (x, y) and the paths P, and P2 have been chosen so that s + t is minimum. Then let S, = {(z, n,) : z E G(u,)) and S2 = {(a,, w) : w E L(u,j}. Since H is a l$ graph, it is clear that (P1 U P2) n (S, U S2)# @ On the other hand, it is clear that JP! n(S, U S,)( G 1 for i = 1,2. In view of the obvious symmetry and duality, we may therefore reduce the remainder of the argument to the following three cases. Only in the third case will we require the additional hypothesis that H admits no -Type 1 exchanges. Case 1. IP,nS,I= I and JP,nS,l= 1. Jn this case, we may assume that (z, &E PI n S,, (z, a,) = (t+,, % ,+1), (a,, w ) E P,f7 St, and (a,, w) = (,u. ,,,, vj,,+l). Recall that PU h$ is a partial order on X. It bollsw~ that z > u2 L>aJ :b w in PU N$, Nhich implies that zl, = x # uz. Therefore, A generalization of Turcin’s theorev to directed graphs l-/L Ps = {(vi, ui+J: 1 d j< jO} is a path in Hfl H’ from x to a*. However, H also contains the path P4 = {(q, h+,) : 1 G i < zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH io}U {(z, a 2 )} from x to + Since (z, a,) E P4 - P3, we conclude that P3# P4 which contradicts the assumption that H is a I_$ graph. Case 2. IPinS,(= 1 and lP2flS,l= 1. Choose io, j. so that (z, a,) = (I+,, %,+I) E PI n S1 and (z’, a,) = (Q,, ui,,+,)E P2 I? Sa. Since s + t is minimum, we must have i. = t and j0 = s, i.e., k(t+i= y = Then it follows that H contains the nonidentical paths P; : %,I = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Q3. {(~,~+~):1~i~t-l}U{(z,u~)} and P~={(Uj,Uj+~):l~j~~-l}U{(Z’,U~)} from s to uZr and therefore x > u2 in P. Since (u2, a,) is an edge in @, we conclude from Lemma 6 that x > y in P, which is a contradiction. Case 3. IP, 17S,l= I and IP2 n (S, U $ )I = 0. Choose i. so that (z, a,) = (q,, r+,+,)~ PI fl S1. Now suppose that i,< r. Then (r.+,+i, I+,+~)= (a,, r+,+2)E H’ implies that (a,, u,,+~)$ N$- &I, i.e., either (a,, 4,,+2)E H or u2 > 4,+2 in P. If (a,, u~,+~)EH, then H contains the nonidentical paths P3={(r.4, 4+i): I siCi,}U {(z, a ,), (a 2 , ~q,+~)}U {(t 4h+,): , i,,+2G i St} and P2 from x to y. On the other hand, if u2 > ho+2 in P, then it follows from Lemma 6 that x > y in P which is a contradiction. We may therefore assume that i. = t and At this stage of the argument, we require that H admits no a3 = Y = ut+1= Vs+l. Type 1 exchanges. Since P2 n (S, U S,) = fl. i.e., P1:E H, we know that s = 1, x = ul, and y = u3 = v2. since PI and P2 are edge disjoint and (2, u3) E PI, we know that xf z and t 32. Therefore t = 2 and P, = {(x, z), (z, a,)}. Furthermore, we know that {(x, z), (z, u2)}s H and since H admits no Type 1 exchanges, we must have x >u2 in P, which in turn implies that x > u3 = y in P. The contradiction completes the proof of this case bnd the theorem as well. Cl We illustrate the preceding theorem with the I$ graphs in Fig. 5. We call the exchange of edges in Theorem 8 a Type 2 exchange. For example. we leave it to the reader to verify that for the poset shown in Fig. 5, the graphs HI a b C d zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB g g h% h% Ii Fig. 5. Ii’ 176 S.B. M aurer, I. Rabinovitch, W .T. Tmfter, Jr. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 5 5 Fig. 6. and ff2 as shown in Fig. 6 are maximal UF graphs which do not admit Type 1 or Type 2 exchanges. and that H, is the unique maximum J.$ graph for this poset. 4. Tbs exfremal problem In this section, we will apply the theory developed in the preceding two sections to determine the maximum number of edges in a U$ graph of a carefully constructed poset (X. P). As a consequence, we will solve the original extremal problem: the determination of I.&I, m). For integers R, m with n ant 22, let X(n, nr) = (X(n.. m), P(n, m)) be the poset defined by X(n. m)={l,2,3.. , ., n} and P(n.m)=((i,j):l~-i<++mj~n}. For example the poset in Figs. 3 and 4 is (after relabeling) X(6,3), and the poset in Figs. 5 and 6 is (after re!abeling) X(8,4). To determine the acyclic digraph of nonforcing pairs for X(n. m ), we use the 1ineEr order L = T, = {(i, j) : 1s i <j =Gn} to break the tics, Thus A$ = ((i, j) ET,, : j 2 I + m}. We then define w(n, m) to be the maximum number of edges in a b? subgraph for the poset X(n, m) and reduce the determination of u(n, m) to the determination of w(n, m). The equivalence of the two problems is easily established by the fo!lowing lemma which is an immediate consequence of the definitions and the fact t.hat IP(n, m)j = V-y +p. Lemma 9. Let n 2 w 32 and (X, P) = X(n. m). Then a subgraph HG N$ is a U$ grupdz if and only if H L1P(n, m) is a m- lot dfy unipathic subgraph of T,. Furthenmre, u(n, m) = w(n, m) + (“- ‘J+‘). Lemma 9 allows us to apply the exchange theorems developed in Section 2 to m-locally unipathrc subgraphs of T,. We will selectively apply these exchange theorems in the proof of the principal theorem, A generalizationof Turbn’s theorem to directed graphs The next lemma establishes some combinatorial in future arguments. 177 identities which we will req=iire Lemma 10.The foIlowing identities hold: (i) h(m, q, r) = h(m - 1, q, r-2)i-(q$2)+(n-q-2)(q+ 1) wi;aen m 23, q%O, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ra 2 , zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC and it=m(ql)+ra3. (ii) h(m,q,r)=h(m-l,q,r-l)+-(q~l)+(n-q--l)q+~~r] when ma3, qa0, r>l, and n=tn(q-l)+rs3. (iii) h(4p+1,q,2p)=h(4p,q-l,6p-2)+(q~‘)+(n7q-l)q w&ten pal, qsl, arzd n=4pq+2p. (iv) h(4p+3,q,2p+1)=h(4p+2.q--1,hp+1)+(q~’)+(n-q-I)q wlien pal, qal. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and n=4pq+2q+2p+l. Proof. To establish the first identity, consider the complete multipartite graph H = H(n2. q, r) having h(m, q, r) edges. Using the notation of Section 1, we label the q + 2 parts of H by Vo, VI, Y2,. . . . V,, V4+1 with IV,l= l$i IVq+,l= f$rl. and IViI=m-l for i=l.2.... , q. Then let S be a q+2 element subset of X,=(1,2,3 ,..., n} chosen so that S contains one element from each of the sets V,,. VI, V,. . . . 1 Vq+,, and let H’ be the restriction of H to X,, - S. Then IH’i = h(m - 1, q, r - 2). Now consider the edges in H-H’. There are (qz2) edges in H - H’ with both endpoints in S, and there are (n -q - 2:l(q + 1) edges in Z -f- H ’ wish one endpoint in S and the other in X,, - S. The! identity follows since H= H’U(H-H’). ‘To estabikk the second identity., we modify the argument given above as follows. We choose a q + 1 element subset S c X,, consisting of one element from each of the se.& V,, V,, . . . , Vq+rr and let I-I’ be the restriction of N to X, - S. Then (f-f’/= h(m - 1, q, r- 1). There are (“;I) edges in H-H’ with both cldpoints in S, there are [$rJ(q + 1) edges in H - H’ with one endpoint in V,, and thiere are (n - L&J-q - 1)q edges in H- H’ with one endpoint in X, - S - V,, and the other in S. The desired identity follows as in the previous paragraph sicce H = H’ U (H - H’). To establish the third identity, we consider a q + 1 element subset of X,, containing exactly one element from VI, V,, . . . , V,,, . Let S fI Vi = (Xi) for Then let Vq=V;UVb where IV$=33p lVbj=p, and xq,Vi. i = I,&. . ..q+l. Let H’ be the restriction of H to X,,, -S. and let H” be the complete q + 1 multipartite graph whose parts are VoU(Vq+, -{x,+~}) U VZV, -{xl}. V,V,_,lx,- ,}, VG- {x,}. Since ~V ,,U (V q+,-{.u,+,})U ~~~=lV ~-~Sq}l:= ix,}, . - . 1 3p-I and \Vi-{~}l=4p-l for i=i,2,...,q-1, we conclude that IH”I= h(4p, q - 1,6p - 2 ). But Z-Yis formed from H’ by adding (3p - 1)p edges between vertices in V; and Vi and deleting 3p2- 2p edges with both endpolints in S.B. M aurer, I. Rabinovitch, W .T. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP T FOt b3 , J r. 178 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA vouwq,,-bq+,w vi. Fran the second identity we have IHI = lW’l+ (4;‘) + (n - q - l)q + p. and since IZ-I’I= I##“\- p, the desired identity follows. The proof of the last identity is similar and is omitted in the interests of brevity. 0 For an edge (i, j) ET,,, we define the length of (i, j) to be j - zyxwvutsrqponmlkjihgfedcbaZY i. Note that each edge in P<r.l.m) has length at least m, and we may therefore view the edges in P(n. m) as “long” edges. Furthermore, if H is a m-locally unipathic subgraph of T, having u(n, m) edges, then P(n, m) E H. On the other hand, there are limitations on the number of “short” edges a m-locally unipathic subgraph of T,, can contain. For example, the restriction of a m-locally unipathic subgraph to a set of m consecutive vertices contains at most l$m” ] edges. ‘Ihe next lemma also limits the number of short edges. Lemma 11. Let n z m 3 2 and let H be a m-bcally unipathic subgraph of T,. Also letibeanintegerwith I<i<i+m1 C n. Theri H contains at most m - 1 eilges from t!re 2m - I element set K{(i,x):i<xGi+m- l}U{(y,i+m-l):iqyCi+m-1). Pro& Suppose first that H contains the edge (i, i + m - 1). Since the restriction of 2-I to the set jji,i+l,i+2,..., i + m - 1) is unipathic, it follows that for each j with i Cj C i +,m - 1, H contains at most one edge from the pair {(i, j), (j, i + m 1). Since there are m - 2 integers between i and i + m - 1, we conclude that H contains at mclst 1 + (m - 2) = m - 1 edges from K. On the other hand, suppose that (i, i + m - 1; +!H. If it is still true that H contains at most one edge from the pair {(i, j), (j, i + m - l)} for each j with i C j < i t m - 1, then it follows that H contains at most m - 2 edges from K. So we may assume that there exists an integer j0 with i zyxwvutsrqponmlkjihgfedcbaZYXWVU C j. C i + m - 1 for which H contains both (i. jO) and (it,, i -+m - 1). Since the restriction of H to {i, i + 1, i + 2 .., i+m1) is unipathic, it follows that for all j with i <jCi+ m - 1 and js jn, H contains at most one edge from the pair {(i, j), (j, i + m - 1)). Therefore IHnKIs2+(m-3)=m-l. El We next introduce a technique for considfering subsets S of X,, for which the identities in Lemma 10 as well as the restriction on the number of short edges given in Lemma 11 will be applicable. This technique will allow us to construct an inductive argument for the principal thc:orem utilizing the following convention. If S s X, and ISI = s with 0 c s c n, then the restriction of T, to X,, - S is isomorphic to T,_,. Given integers m,, m2 with n am, 22 and n--s> m,a2, we may consider a m,-locally unipathic subgraph H of T, and its restriction H’ to X, - S. We may then ask whether H’ is a mz-locally unipathic subgraph of T,_,. For integers n, m, k with n am 2~2 and Oskkm, we let S(n,m, k)= {iEX, :i = k (mod m - l)] and s(n, m, k) = IS(n, m, k)l. A generalizationof Turcin’s theorem to directed graphs If9 L4?mma 1% . Let n, m, k be integers with n > m > 3 and 0 G k < m, and let H be a m- locally unipathic subgraph of T,,. If H’ is the restriction of H to X, - S(n, m, k) and s = s(n, m, k), then H’ is a m - l-locally unipathic subgraph of zyxwvutsrqponm T,,-,. Proof. Let A be a set of m - 1 vertices which occur consecutively in ‘I’“__%. If the vertices in A also occur consecutively in T,, then since H is m-locally unipathic, it is also m - l-locally unipathic, and the restriction of H’ to A, which is the same as the restriction of H to A, must be unipathic. Qn the other hand, if the- vertices in A do not occur consecutively in T,,, then it follows that there is a unique element x E S(n, m, k) so that A U(x) is a set. of m consecutive integers in T,,. As before. the restriction of H’ to A must be unipathic, and the argument is complete. Cl We pause to detail two exceptional cases. The following mediately from the remarks at the end of Theorem 2. result hollows im- Lemma zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 13. u(3,3) = 2. Furthermore, there are exactly three 3- focally unipathic subgraphs zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of T3 which have two edges: H(3, (1,3) ={(l, 2), (1,3)}, ri(3,(), 3) = {(1,3), (2, 3)}, Ho = {(1,2), (293)). We next discuss the special case (n, m) = (5,4). The argument will be generalized to obtain the principal theorem. presented here Lemma 14. u(5,4) = 7. Furthermore, H(4, 1,2) is the unique 4- local!g unipathit s&graph of T5 having 7 edges. Proof. Let H be a 4-locally unipathic subgraph of T5 with 1HI = u(5,4). Then let S1 = S(5,5,1) = {1,4}, S2 = S(5,4,2) = (2,5}, s1 = IS,( = 2, and s2 = IS,,1= 2. Also let H, denote the restriction of H to X,-- Si and let & = (H- Hi) n P(5.4) = {(1,5)} for i = 1,2. Then set El ={(4, 5))fl H and E2={(l, 2))rl H. Finally, let Ii=Hi- Li- Ei for i=l,2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA N OW Hi is a 3-locally unipathic subgraph of T3 SO IHi(G 2 for i = 1,2. Also, we note that I&IG 1 for i = 1,2. We next show that /Ii U I +1=~4 for i = 1,2. However. this follows immediately since l&l = 1 and JIiJ65 by Lemma 11 for i = 1,2. 7C~(5,4)=jHj=)H,l+(E,I+I1~U&~~2+1+4=7. Thus IHI= Therefore, u(5,4) = 7. We now proceed to show that H = H(4, 1,2)1. We begin by observing that we must have lHil = 2, jEil = 1, and I~iU Lil = 4 for i = 1.2. In particular, we knov that Ii1 and Hz must be one of the three extremal graphs in Lemma 13, and we know that {(1,2), (4,5), (1,5)}~ H. Suppose first that H, = H(3,0,3), i.e., ((2. 3), (2,5)}c H. If H contains the edge (3,4), then H contains nonidentical paths from 1 to 4. The contradiction requires 130 S.B. kiaurer, I. Rabinovitch, W.T. Trotter, Jr. that (3,4)&H. This in turn implies that H2# fi(3,0,3) and H2# Ho. i.e., Hz= H(3,0,3). Thus zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA H contains the edges (1,3) and (1,4) which in turn implies that H contains nonidentical p;a;>s from 1 to 3. The contradiction shows that H1# .H(3,0,3). At this point, we may take advantage of the obvious duality to conclude that H& &3,0,3). (3,5)}. Then we observe that Hz= Now suppose that H, = H,={(2,3), H(3,0,3j = {( 1,3), (1,4)], and H2 = Ho = {( 1,3), (3,4)) imply that H contains the edge (1,3). This in turn implies that H contains nonidentical paths from 1 to 3. The contradiction shows th.ai H, # Ho, and by duality, we may conclude Hz+ Ho. Therefore H, = fi(3,0,3) arid Hz = H(3,0,3). But these statements imply H = H(4, 1,2). Cl The next lemma allows us to resfrict our attention to m-locally unipathic subgraphs which do not admit Type 1 or Type 2 exchanges. This will simplify subsequent arguments consideraly. EABUIM15. Let n, m be integers with n zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG 3 m ~3. If q and r are integers for which n=q(m1)4- r. (m, q, r)f (3,0,3), then - l)] , and M m - 1>‘Is r S [$(rn H( m. q, r) and fi(m, q, r) cannot be obtained from a m- locally unipathic subgraph H of ‘I’, by an exchange of Type P or Type 2. first that H(m, q, r) can be obtained by a m-locally unipathic ?& by a T;ype 1 exchange. The argument for &m, q, r) is dual. x, y, z wit!h 1 GX C y < z 6 x + m - 1 for which R contains (x, z) one of these edges is exchanged for (x, z) to form H(m, q, r). Without loss of generality dH(m, q, r) = (H - {(x, y )}) U {(x, 2)). Choose an integer i so that x E Vi. Since (x. y) $ H(rit, q, r), we know that y also belongs to Vi. Since z-xsnd1, we know that ZE Vi+,. If i >I), let w denote the largest integer in Vi _,. Then him, q, r) and N contain the edges (w, x) and (w, y). But H also contains (x, y) which is a contradiction since y - w G m - 1. Therefore i = 0, r 2 3, and m a 6. If IV,1 22, we may consider the first two integers in V, and choose one of them, say z’, with z’# z. It follows that H contains (x, y) (y, z’), and (x, 2’). Since z’a2+t$rj and x21. we see that r’- xsl+[% rJsm- 1 which is a contradiction. Now suppose that H(m, q, ri is obtaintd from a m-locally unipathic stibgraph H of T, by a Type 2 exchange. Choose ijltergers x, y, z, w with 1 <x < y < z < w < x + m - 1 for which (x, y)~ H, (z, W )E H, (x, z)$ H, (y , t)+! H, and zyxwvutsrqponmlkjihgfedcb H(m, q, r) is then obtained from H by a Type 2 exchange which results in (x, y) being exchanged for (x, z) and (z, w) being exchanged for (y, w). (Other exchanges may also be involved but this will not matter.) We then choose an integer i for which X, y E Vi and Z, w E Vi+l. Hence H contains both (x, w) and (y, z) but this implies that N contains nonidentical paths from x to w. The contradiction co!mpletes the proof. 0 Pro& Suppose subgraph H of Choose integers and (y, z) but A generalization of Turdn’s theorem to directed graphs We are now ready to present the principal theorem Theorem 181 of this paper. 16. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Let n 2 m 2 2. Then the maximum number u(n, m) of edges in a m- locally unipathic subgraph of T,, is zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM h(m, q, r) w here q and r are the unique integers satisfy ing n = q(m - 1) + r and r&n - 1)1 s r C [s(rn - l)]. fiurthermore: (i) If m = 2, then T, itself is the unique m- locally unipathic subgraph of T, having u(n, m) edges. (ii) If n = m = 3, then there are three m- Iocaily unipathic subgraphs of ‘T,,having u(n, m) edges: H(3,0,3), &3,0,3), and Ho = {(1,2), (2,3)}. (iii) If n 3 m > 3, r >i(rn - l), and H is a m- locally unipathic subgraph of T, having u(n, m) edges, then either H = H(m, q, r) or H = fi(m, q, r). (iv) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA If nam23, r=$(m-l), (n, m)Z(3,3), and I-I is a m- locally unipathic subgraph of T, having u(n, m) edges, then either H = H(m, q, r), H = fi(m, q, r), H=(m,q- l,r+m- 1), orH=fi(m,q- l,r+m- 1). Proof. We first dispense of the case m = 2. In this case, we observe that ‘I’, itself is the only 2-locally unipathic subgraph of T, having u(n, 2) = (;) edges, and the desired result follows since T, = H(2, n - 1, I). We may also assume (n, m ) # (393). We then assume validity for all values of m with m d p where p is some integer with p 2 2 and consider the case m = p f 1. In view of Theorem 2, we may assume n > m. Throughout the remainder of the argument, q and r will denote the unique integers for which n = q(m - l)+ r and [i(rn - l)l Qr < [$(m - 1)l. From this point on, we proceed with :an indirect proof. We assume that the theorem is false and let ie denote the set of all counterexamples, i.e., U: is the set of all m-locally unipathic subgraphs of T, having u(n, m) edges other than the canonical graphs given in the statement of the theorem. We may then choose a counterexample HE % which does not admit either a Type 1 or Type 2 exchange. To see that this is possible, we observe that each time an exchange of either Type 1 or Type 2 is performed, the sum of the lengths of the edges in the graph increases, but of course the number of edges remains the same. On the other hand, it follows that if we choose a graph HE % for which the sum of the lengths of the edges in H is maximum, then H does not admit either a Type 1 or Type 2 exchange. Otherwise, the exchange would necessarily transform H into one of the canonical extremal graphs which is impossible by Lemma 15. It is important to note that the counterexample H satisfies the following two properties. G m, (x, y) E H, and (w, z) E H, then w ( y. P,: If lsx<y<z pz: If n-m+l<x<y<z, (y,z)~H, and (x. w)EH, then y<w. We first establish P,. Suppose to the contrary that 1 s x -Cy < z < m, (x, y) E H. (w , Z) and y G w. Suppose first that y = w. Then H contains (x, y) and (y, z) and admits a Type 1 exchange. Wow suppose y < w. If H contains either (x, wj or (y, z), it admits a Type 1 exchange and if H contains neither (x, w) or (y. z). then S.B. Maurer: I. Rabinwitch, 182 W. ‘p: Trafter, Jr. it admits a Type 2 exchange. This, completes the proof of P1. The proof of zyxwvutsrqponmlk P2 is dual and is therefore omitkd. At this point, we divide the remainder of the argument into four cases zyxwvutsrqpo depending on the magnitude of r Case 1. m SrC [$(m-1)l. Let S,=S(n,m, l), &=S(n,m, l+ E(r-m)]), S3=S(*r, m,r-m+l), and&= S(n, m, 1+ [JrJ). Note that IS,1 = IS.1 = IS,! = q+ 2 and IS,,1= q + 1. For convenience, we also let q = lS,l for i = 1, :‘_,3,4. Then for i = 1,2,3,4, let Hi be the restriction of H to X,, - Si. It follows from Lemma that Hi is a m - l-locally unipathic subgraph of Tnmg-2 for i = 1,2,3, and that H4 is a m - l-locally unipathic suhgraph of T, - q - 1. We next observe that the equation st = q(m - 1) + r, and the inequality m G r < p$(rn- :;I together imply that the following statements hold: (a) n-q--2=q(m-2)+r-2 and [$(m-2)1cr-2<[~(m-2)1. l=q(m-2)+r-1 and (b) If rf4(3m-4), then n-qkl(rn - 211. (c) If r=&(3m-4), then n-q-l=(q+l)(m-2)+r-m+l) $(m - 2). [$(m-2)1sr-l< and r-m+l= It follows from the inductive hypothesis that /HiI G u(n -q - 2, m - 1) = h(m-l,q,r-2) tor i= 1,2,3. If r# i(.3m-4), then lH4jGr&z-q-l,m-1)= h(m-l,q,r-1). On the other hand, if r = &3m -4), then j~-p,IG u(n-q-1,m-1)=h(m-1,q+1,r-rn+1). But since r-m+l=j(m-2), we have ~H~~~h(m-l,q+1,r-m+1)=k(m-l,q.r-1). We conclude that IHJ< h(m - 1, q, r- 1) for ail values of r treated in this case, We now describe a method for partitioning each of the sets H-Hi into three subsets. First, we let Z+= (H-Hi) 17P(n, m) for i = 1,2,3,4. Then let ai be the least integer in Si and bi the greatest integer in Si. We define Ei = H n {{(x, Cli): 1 d x < 4) U {(bi, y) : bi < y s n}) fori=1,2,3,4.Finally,wesetZi=H-Hi-&-Eifor i=l,2,3,4.(Weusethe letters C, E, and Z to suggest “long”, “exterior”, “interior” respectively.) We now proceed to examine the number of edges in these sets. First, it is easy to see that Z+ contains @)- (Si - 1) edges with both endpoints in Si. If x E X,, - Si and a, < x < bi, then there are Si - 2 edges in Z+ having x as one of its endpoints. If x E X, - Si and either x < ai or bi < x, then there are si - 1 edges in & having x as one of its endpoints. Therefore, ss IL-() .2 I- ,, -(sl - I)+(& - l)(m -2)(sie2) +[n-(Si-l)(m-l)-l](si-1). Secr;nd, we observe (Si-I !m-1). that it follows immediately from Lemma 11 that IZila A gerreralization of Turh’s theorem to directed graphs 183 We conclude that = 0z +(tl-Si)(Si- 1). The form of the preceding inequality is not surprising since it is immediate that 14ULil=(S;)+(n-si)(si-l) if H=h(ffl,q,r). In this case, note that H contains each of the (“1)edges with both endpoints in Si, and if x E X, - Si, then there are Si- 1 edges of H- ,Ei -Hi joining x with a point Of Si. We may combine these inequalities with the identities in Lemma 10 to obtain the following inequalities. ch(m--l,q,r-2)+ sh(m-l,q,r-q)+ Gh(m,q,r)+JEiJ z zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON +(U-Si)(,Sil)+(Ei] 0 ( for “i’ ) +(n-q_2)(q+ 1)+1&l i=l,2,3. sh(m-l,q,r-l)+(;)+(n-s4)(s4-l)+lE4( s Mm, q, r) + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED lE41- kl . We conclude from this that we must have lE4l > [$rJ. Now suppose that IEiI> 0 for i-1,2,3. We not? that a,=l, az=l+I_$(r-m)], a3-l-tr-m, a4=1+[iaJ. b,= n-r+m,I;~,=n-~~(r-m)l,b3=n,andb,=n-~~r~+l.SincelE,l>Oand(E,(~ 0, we know that H contains an edge e, = (n - r + rn, j) where n - r + m < j G n and m. Since l&j > 0, we know that either edge e,=(i, l+r-m) where 1 < i < 1 + r - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC H contains an edge e2 = (i’, 1+ [;(r - m)J) where 1 G i’< zyxwvutsrqponmlkjihgfedcbaZYXW 1 + @r - m)J or an edge e; = (n - [$(r - m)J, j’) where n - [$(r - m)l <j’s n. Now suppose that H contains an edge e2 = (i’, 1+ li(r- m)]). Since H satisfies property P,, it follows that if (x, QJ E Ed, then l<x < 1 +$(r- m). Similarly, since H satisfies P2, it zyxwvutsrqponmlkjihg follows then n-r+ m <yin. It follows that IE4Is that if (b,, Y)EE~, 14(r- m)J + r - m &r - m)J which is impossible since 1E.J2 l.$rJ and [ir] > [$(r - m)l . On the other hand, if H contains an edge es = (n - [i(r - rn 11, j’) then 184 S.B. Maurer. I. Rabinouitch, W.T. Trotter,Jr. we would coslclude that if (x, a,) E &, then 1 =Gx < 1+ r - m, and if (h,, Y) E Ed, then n- [$(r-m)l <y<n. 1, follows that lE41d(r-m)+ [&r-m>] = [s(r-m)l. As before, this is impossible since zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH [EdI2 [$rl and [&‘I > @(r - m>lThe contradiction allows us to conclude that there must be some i E {1,2,3} for -2), we note that this implies in turn that which l&l= 0. Since r-2#i(m IH;=u(n,m)=h(m,q,r), IHJ=h(m-l,q,r-2), IVJ41=(q;2)+(~-q-2)x (q+l) and IL;I=(q+l)(m - 1). Therefore, either Hi = H(m - 1, q, r - 2) or Hi = fi(m - 1, q, r- 2). In either case, it is easy to see that Hi contains q + 1 edges of length one. Furthermore, if we choose an arbitrary consecutive pair ul, USE Si, then there exists a unique edge (w, w + 1) E Hi zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO SO that 1 s ul C w < w + zyxwvutsrqponmlkjih 1 < 02 = tr, + nt c n. Sincz !fil = (q+ l)(m - l), it follows that H contains exactly m - 1 edges from {(u,. x):u~<:x~~)~}U{(~, u~):zJ~~~<u~}. Thus if zll <x<u2, then H must contain at least one of (u,, x) and (x. u,). First, suppose that uI <x G w. We show that (x, u2) E H. To the contrary, assume (x,~;~)$H; then (u,,x)~H. Now (x,w +l), (w,w+l)$?HinH so (ul,w)$- H, (w , u2jE H, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ii.e ., H contains (w, w + l), (w + 1, u2), and (w, u2) which is a contradiction, We conclude that if u, <x < w, then (x, u2) E H. A dual argument shows that if w + 1 G y < u2, then (Q, y) E .H. We now show that H contains (u,, uz). To the contrary, suppose that (ul, u2)$ H. Then there exists an integer x with u, <x < u2 for which H contains both (u,, x) and (x, 21~)~ If x d w, then H oDntains (u,, x), (x, w + 1), and (ul, w + 1) which is a contradiction. Similarly. if w -+ 1 G x, then H contains (w. x), (x, u2), and (w, u2) which is also a contradiction, W;= conclude that (ul, u2j E H. In the above argument. ul and u2 were an arbitrary consecutive pair from Si so that we have determined the location of each of the (q+ l)(m - 1) edges in IiSince Ei = (4 and ti G P(n. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA m)~ H, it follows that if Hi = H(m - 1,~. r‘-2), then H = H( m q, I), and if ~ii = fi(m - 1, q, r - 2), then H = d(m, q, r). Of course, we have obtained a contradiction since the assumption that H was a counterexample has led to the conclusion that H was not a counterexample. With this observation, the proof of Case 1 is complete. Case 2. &m - l)<:rCm. In view of Lemma 14, we assume (n, m)# (5,4). Consider the three sets Sl= S(PY,m, 11, Sz= S(n, m, r&l), and S,= S(n, m, r). I&t Si = ISiI and let a, and bi denote the least integer and the greatest integer in Si respectively for i = 1.2,3. Note that Si = q + 1 for i = 1,2,3. We then define for each i = 1,2.3 the subgraphs Hi* Ii, &,, and E, exactly as in Case 1. Since [s( m - 2)1 < r - zyxwvutsrqponmlkjihgfed 1 < [s(m - 2)1, we know that the following inequality holds. Sh(r?~-l,q,r- = h (rn. q , rlf l)+ (“5 *)+(n-q- l)q+lE*l IEij- Lir.1 for i = 1,2, 3. A generalization zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC of Turh’s zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF theorem to directed graphs 185 In particular, we note that lEil> [$rj for i = 1, 2,3. If IF, I= [$rJ for some zyxwvutsrqp then we know that H,=H(m-l,q,r-I) or Hi-=~~(m-l,q.r-1) unless r - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1 = $(m - 2), in which case, we may also have Hi == H(m-l,q-l,r-m-3)orHi=~(m-l.q-l,r+m-3).However,itiseasyto show that the requirement that l&l 3 l$rJ for i = 1,2,3 rules out this possibility. To see that this is true, we observe that if Hi = H(nz - 1, q - 1, r + w - 3) or Hi=A(nl-l-q-1, r+m-3) for so.me ifz{1,2,3}, then Hi contains an edge (y,y+l) where either y= [i(r+m-1)j or y= [&r+m-1)l. Since IE,[->O, H contains an edge of the form (x, r) where 1 %x <r. But this implies that H violates Property P2, since l<x<r<y<y+ 1 G ml. We may therefore assume that either Hi=H(m-l,q,r-1) or Hi= Z$(m-l,q,r-1) whenever IEil=lsr] and iE iE(1,2,3} , U.2,3} . Suppose first that IElI= [$rJ. Th enwemusthaveH,=H(rn-l,q.r-l).forifr is even and H1 = fi(rn - 1, q, r -. l), then H, contains each of the $r - 1 edges in the set {(n-$r+ 1, x):n-$r+l<x G n). However, this implies that jE,l G ir - 1 < [$J which is a contradiction. Since H, = H(m - 1, q, r - 1). we know that H contains each of the l$r] edges in the set {(n - [$rj, x): n - [$r] <x c n} , and thus El = {(n - r + 1, x) : n - [ir] <x d n} . The argument in Case 1 may now be applied to determine the edges in Ii and show that H = H(m, q, r). We may therefore assume that lE,l> Ur]. Dually, we may assume that lEJ> [$rJ. It follows that E, contains no edges from the set {(x, [&rl): I G x s [lrl}. for otherwis: we would conclude that lE.ll~ l$rJ. Therefore Ez = Lhr] <x s n} and lEz\ = @r] . However, this in turn requires that {(n- lar],x):nlEl/ S l$rJ which is a contradiction. This comple[es the p,roof for Case 2. Case 3. $(m - 1) = r and r is even. First set r = 2p and m = 4p + 1, We then consider the sets S,, Sz. . . . . S,. where Si = S(n, m, i) and .si= q + 1 for i = 1,2, . . . , r. Note that U(n-q-l,n&-l)=u(4pq+2q-q-l,4p)=h(4p,q-l,hp-2). It follows that if It; =O for some ie{I,2,...,r} . then Hi=H(4p,q-1.6p-2) and the same argumeiat used in Case 1 wouid a:!ow us to conclude that H=H(m,q--1, r+m- l)= H(4p+ 1, q- 1,6pl. We may therefore assume that tEi I > 0 for i =: 1,2, . . . , r. Now consider the set Sr+, = S(n, m, 3~). Since s,,, = q and u(n - 4, m - 1) = u(417q + 2p - q, 4~) = h(4p, q, 2p), we conclude that /Er4,j22p. If IEr+,l = 21.4it follows easily H(-? + 1, q, 2~). We therefore assume that (E,+,j > r. Next suppose that for some i E { 1,2, . . , r} , $ contains lsx<i and an edge (n-r+i,y) where n-r+i<ySn. clude that jE,+,Is(i - l)+ r-i = r - 1. The contradiction that H = H(m, q. r) = an edge (x, i) where Then we would conshows that for each S.B. Maurer, I. Rubimuitch, 186 i= 1,2,3,. . . . r, we have either W.T. Trotter, Jr. Eic{(x,i):lGx<i) or zyxwvutsrqponmlk $S {(n-r+i,y):n-r+iCysn} . Similar reason& shows that if H contains an edge of the form (x, i + 1) where and an sdge of the form (n-r+i,y) where n-r+i<ySn, then \&+,I c r. The contradiction shows that we must either have 4 c {(x, i): 1 G i C i} for i= 1,2,. . . , r, neither for i = 1,2,. . . , r or EiC((y, n-r+i):n-r+i<yGn} of which is possible. The contradiction completes the proof of this case. Case 4. #m - 1) = 4, r is odd, (n, m)f (3,3). First set r = 2p + 1 and m = 4p + 3. As in Case 3, we consider the sets S,, s,, . . . , S, where Si = S(n, m, i) and Si = q + 1 for i = 1, 2, . . . , r. Note that IbEx%i ~(n-q-l,m-1)=u(4pq+2p+q,4p-t-2)=h(4pt2,q-l,6pt1). lt follows that if I$! - 0 for some i E { 1,2, . . . , r}, then either Hi=Zf(4p+2,q-l,6p+l) or Hi=H(4p+2,q-l,6r+l). Applying &heargument used in the previous cases, we would conclude that either H=H(4p+3,q-1,6pt3) or H=fi(4p+3,q-1,6pt 3). We therefore assume IEi1>O for i = 1,2, . . . , r. Mow consider the set S,+, = S(n, m, 3~). Since s,,~ = 4 and u(n-q,m-l)=u(4pqt2p+qtl,4p+2)=h(4p+2,q,2p+l), we conclude that lE+,] ar. If [&+,I = r, then it follows easily that H=H(4p+3,4,2ptl) or H=fi(4~+3,q,2p+l). We may therefore assume that IE,+,I > r. The remainder of the case follows along the same lines as Case 3 and is therefore omitted. With this observation, the proof Cl of our theorem is complete. 5. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The compu4utb of rank zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA If P and Q are par&i orders on a set X and P cc0, we say that Q is an extension of P. If Q is also a linear order, then we say Q is a linear extension. A well known theorem of Szpilrajn [7] asserts that if P is a partial order on a set X, then the collection 9 of all linear extenl;ions of P is nonempty and n 9 = P. A family 9 of linear extensions of a partial order P is called a realizer of P when n 9 = P. A realizer 5 of P is said to be irredundantwhen n .%# P for every proper subfamily $5 5 Dushnik and Miller [l] defined the dimension of a poset (X, P) a% the smallest integer t for which there exists a realizer 9= CL,. L*, . . , &) of P. Note that if (X, I?) Slas dimension t and 9 = {L,, L2, . . . , I;} is a real,zer of f, then 9 is irredundant. Maurer and Rabinovit8zh [2] defined the rank 4 tX, P) as the largest integer t for tl*lhich,there exists an irredundant realizer A generalizationof Turcin’s theoremto directed graphs 187 9=(L1,L2,..., &} of P and showed that wbile a n-element antichain has dimension two when n a 2, it has rank l&z’] when n a4. In [6], Rabinovitch and Rival gave a formula for the rank of a distributive lattice. In [3] and [4], Maurer, Rabinovitch, and Trotter developed a general theory of rank based on the graph theoretic concepts discussed in Section 2 of this paper. For the sake of completeness, we state here the principal results of this theory. For n 20, let n and A denote respectively an n-element chain and antichain. If X = (X, P) and Y = (!Y, Q) are posets, we define X join F, denoted XCBY, as the poset (X U Y, P U Q IJ (XX Y)), i.e., in XCDY, every element of X is greater than every element of Y. A poset (X, P) is said to be rank degenerate if there exist integers n, m a0 such that (X, P) is isomorphic to a subposet of n@?@lrlz. The width of a poset (X, P) is the maximum number of points in an antichain contained in (X, P). Theorem 17 [5]. If (X, P) is rank degenerate, then rank(X, P) = width(X, P). Theorem 18 [3]. If (X, P) is not rank degenerate, then the rank of (X, P) equals the maximum number of edges in a V$ subgruph of N$ By combining Theorem 18 and Lemma 9, we can now compute the rank of the family of posets {X(n, m) : n > m >2}. Note that X(n, n) = ii for n 3 2 so rank X(2,2) = 2, rank X(3,3) = 3, and rank X(n, n) = [$n’] when n >4. Corollary 19. Let n > m 22. Then rank X(n, m) = h(m, q, r)- (“-?) where n = (m - 1)q + r and r&m - 1)1 c r < [$(m - l)] I Proof. Note first that X(n, m) is not rank degenerate when n > m so that by Theorem 18, the rank of X(n, m) equals w(n, m), the maximum number of edges in a U$ subgraph of @. In view of Lemma 9, we know that w(n, m)= u(n, m)- i, x-m++ 2 ) and our conclusion follows from Theorem , 16 since u(n, m) ‘=h(m, q, r). CJ It is of particular interest to consider the slpecial case of the preceding result which occurs when n = 2m. The family {X(2m, m) : m 2 1) is a collection of posets of height one of particular combinatorial interest. First, the posets are interval orders of height one and secondly, X(2nr, m) is the horizontal split of m (see [8] S.B. Maurer. 1. Rabinovitch, W.T. Trotter, Jr. 188 for definitions). Corollary X(2m. m) has dimension two for all m zs2, and we may examine 19 in detail to obtain a formula for the rank of X(2m, m). (i) rank X(4.2) = 3. (ii) rank X(6,3) = 7. (iii) rank X(8,4) = 12. (iv) rank X(2m, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA rnj = [$(3m*-3)J for m 25. Proof. X(2m. m) is not rank degenerate when m ~2 so that rank X(2m, m) = w(2m, m),the maximum number of edges in a U$ subgraph of I’$. By Lemma 9, we know that w(2m.m)= u(2m. m)It follows from Theorem 1 ( ) 111+ 2 * 16 that u(4,2j = h(2,2,2) = 6 so tiat w(4,2) = u(4,2) - (if’) = 6- 3 = 3. and w&4)-(;)=h(4,2.2)-(:)-22-M= On the other hand, when m a5, w(2m. m j = h(m, 1, m f 1)- 12. u(2m, m) = h(m,1,m + 1) so that m + 1‘ (,I , = (m - l)(m + l)+ @m + 1yj - (“; = [$3m2- 3jJ. ‘) cl Although we d:o not dtscuss the detaiLi here, it is relatively easy to establish the inequality rank X(2m, m) 2 # rn* - 3)J directly from the definition of rank. This is accomplished by explicitly constructing an irredundant realizer 9 for X(2m, m) with pS[ = [d(3m2-3j]. The problem of establishing the reverse inequality, rank Xt2m, m) s Li(3rn *- 3)J, served as; the initial motivating force behind this PaPer. A generalization of Tush’s theorem to directed graphs 189 6. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Open probllems One of the obvious problems remaining to be solved is to investigate further the relationship between u(n, nt), the maximum number of edges in an m-locally unipathic subgraph of T,,, and A(n, m), the maximum number of edges in an m-locally triangle free subgraph of T,. We recall that A(m, nz) a u(n, m) for all n am 22 and that A(n, n)= u(n, n)= [$“J while b(n, 2)= u(n, 2)=(T). On the other hand, it fnay happen that A(n, m)> u(n, m). For example, when n = 9 and m = 8, ~(9,s) = 20 and the only extremal graphs are the complete bipartite graphs H&O, 9) and A(& 0.9). However, it is straightforward to shalw that A(9.8) = 2 1 andthat{(i,j):1~i~4,5~jc8}U{(j,9):5~~~8}U{(1,9))isanextremalgraph. Several problems involving the digraphs of nonforcing pairs also arise naturally. (1) What (acyclic) digraphs are the (acyclic) digraphs of nonforcing pairs of a poset? (2) Characterize maximal and maxirnum Us graphs. (3) If IX]= n, characterize the set S of integers for which there exists a poset (X. P) so that for every s E S, there exists a maximal V$ graph having s edges. (4) Which posets have the property that every maximum LJg graph admits no Type 1 or Type 2 exchanges. References [I] B. Dushnik and E. Miller, Partially ordered sets, Amer. J. Math. 63 (1941) 600-610. [2] S. B. Maurer and I. Rabinovitch, Large minimal realizers of a partial order. Proc. AMS 66 (107X) 21 I-216. [3] S.R. Maurer. 1. Raoinovitch, and W.T. Trotter, Jr., Large minimal realizers of a partial order Il. Discrete Math, to appear. [4] S.B. Maurer. 1. Rabinovitch. and W. T. Trotter, Jr.. Partially ordered sets with equal rank and dimension, Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Florida, 1979, to appear. [.S] S.B. Maurer, I. Rabinovitch, and W.T. Trotter, Jr.. Rank degenerate partially ordered sets. Proceedings of the Tenth Southeastern Conference on Combinatorics. Graph Theory and Computing, Boca Raton, Florida. 1979, to appear. [6] I. Rabinovitch and 1. Rival, The rank of distributive lattice, Discrete Math. 25 (1070) 275-270. [7] E. Szpilrajn. Sur I’extension de I’ordre partiel, Fund. Math, l/1 (1930) 386-380. [x] ‘W.T. Trotter, .llr. and J.I. Moore. Characterization problems for graphs. partially ordered sets, lattices. and families of sets, Discrete Math. 16 (1076) 36 1-38 I. [9] P. T&m, An extremal problem in graph theory, Math. Fiz. Lapok. 48 (1941) S36-552.