Acyclic orientations of graphs

Let G be a finite graph with p vertices and x its chromatic polynomial. A combinatorial interpretation is given to the positive integer (-l)px(-A), where h is a positive integer, in terms of acyclic orientations of G. In particular, (-l)Px(-1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in the sense of Doubilet-Rota-Stanley, is constructed which yields the generating functions which occur in the above context. 8. The chromatic polynomial with negative arguments Let G be a finite graph, which we assume to be without loops or multipk edges. Let V = V(G) denote the set of vertices of G and X = X(G) the set of edges. An edge e E X is thought of as an unordered pair {u, u) of two distinct vertices. The integers p and q denote the cardinalities of V and X, respectively. An orientation of G is an assignment of a direction to each edge {u, u), denoted by u + u or u + u, as the case may be. An orientation of G is said to be acyck if it has no directed cycles. Let x (X) = x(G, X) denote the chromatic polynomial of G evaluated at X E C. If X is a non-negative integer, then x(X) has the following rather unorthodox interpretation. Proposition 1 A. x(X) is equal to the number of pairs (u, (I), where u is any map 0: I/+ {1,2,..., A) and 0 is an orientation of G, subject to the two conditions: . (a) The orientation 0 is acyclic. (b) If u + u in the orientc:!ion 0, then (T(U) > (J(U). * The research was supported by a Miller Research Fellowship. 172! RX Starde;~, Acyclic orientations of graphs hoof. Condition (8) forces the map o to be a proper coloring (i.e., if {u, U} E X, then O(U) # o(u)). From (b), condition (a) follows automatically. Conversely, if o is proper, then (b) defines a unique acyclic orientation of G. Hence, the number of ahowed u is just the number of proper colorings of G with the colors 1,2 , (.., X., which by definition is x(A). Proposition 1.1 suggests the following mo~dification of x(X j. If h is a non-negative integer, define E(X) to be tile number of pairs (CJ, O), where 0 is any map 0 : I/+ { 1,2, . . . . A} and 0 is an orientation of G, subject to the two conditions:; (a’) The orientation 0 is acyclic, (b’) If u + u in the orientaticn 0, then O(U) > u(v). We then say that B is con-pztible with 0. The relationship between x and z is somewhat analogous to the relationship between combinations of y1 things taken k at a time without repetition, enumeratt 3 by (i) T and with repetition, enumerated by ( n+;-1) x (_l)k(;“). Theorem 1.2. Fo,r all non-negative integers h, X(h) = (I)P x(--X). Proof. Recall the well-known fact that the chromatic polynomial x(G, X) is uniquely determined by the three conditions: (i) x(C,, A) =: X, where GO is the one-vertex graph. (ii) x(G + H, h) = x(G, h) x(H, A), where G + H’ is the disjoint union of G and H, (iii) for all e E X, x(6, X) = x(G\,e, X) -xl(G/e, X), where G\e denotes G with the edge r? deleted and G/e denotes G with the edge e contracted to a point. Hence, it suffices to prove the fo!lowing three properties of g: (i’) g(GO, X) = X, where G, is the one-vertex graph, (ii’) F(G + H, X) = z(G, X) ?(H, A), 5 iii’) R(G, A) = 2(G\e, h) + x(G/e, A). Properties (i’) and (ii’) are obvious, so we need only prove (iii’). Let o’ : V(G\e) -+ ( 1,2, ..,, X} and let 0 :be an acyclic: orientation of G\,e compatible with 0, where e = {u, v} E X. Let Or be the orientation of G obtained by adjoining u + u to 0, and O2 that obtained by adjoining u + u. Observe that o is defined on V(G) since V(G) = V(G\e). We will 1. The chrou $ztic polynomial with negative arguments 173 show that for each pair ((T, Cl), exactly one of Or and O2 is an acyclic orientation compatible with (T, except for X(G/ e, X) of these pairs, in which case both Or and O2 are acyclic orientations compatil;>e with 0. It then follows that 2(G, X) = F(G\e, X) + x(G/e, A), so prcviilg the theorem. For each pair (0, O), where (T: G\e + ( 1,2, . . . . X} and 0 is a:~ acyclic orientation of G\e compatible with O, one of the following three possibilities must hold. Case I : a(u) > o(u). Clearly O2 is not compatible with u while O1 is compatible. Moreover, ill is acyclic, since if u 3 u + w 1 + w2 + . . . + u were a directed cycle in 0 r , we would have II (a j 12 o (u) 2 o(w 1 > 2 U(W*) 2 . . . > U(U), which is impossible. Case 2: U(U) < u(u). Then symmetrically to Case 1, O2 is acyclic and compatible with u, while Or is not compatible. Case 3: u@j = u(v). Both Or and O2 rue compatible with u. We claim that at least one of them is acyclic. Suppose not. Then Or contains a directed cycle u + u + w1 -+ w2 + . . . + u while O2 contains a directed cycleu+u+w~+w)2+ . . . -+ u. Hence, (1 contains the directed cycle contradicting the assumption that 0 is acyclic. It remains to prove that both Ol and O2 are acyclic for exactly y(G/e, A) pairs (u, O), with u(u) = u(u). To do this we define 2 bijection Q,(u, 0) = (o’, 0’) between those pairs (a, 0) such that both Or and O2 are acyclic (with u(u) = u(u)) and those pairs (u’, 0’) such that u’ : G/e + { 1,2, . . . . X) and 0’ is an acyclic orientation of G/c compatible with u’. Let z be the vertex of G/e obtained by identifying ti and u, SO JWIe) = UG\e) -{u, U} u .: Z} and X(G/e) = X(G\e). Given (u, O), define u’ by u’(w) = u(w) for 211 w .s V(G\e) (z) and d(z) = u(u) = u( 3). Define 0’ by w1 + w2 in 0’ if and only if wl + w2 in 0. It is easily seen that the map @(a, 0) = (tr’, 0’) establishles the d.esired bijection, and we are through. Theorem 1.2 provides a combinatorial interpretation of the positive integer I(--1)P ;K( G, -A), where X is a positive integer. In particular, w1len X = 1 evi:ry orientation of G is automatically compatible with every map u: G -+ { 11. We thus obtain the following corolllary. 174 R.P Stml+~, Acyclic orientcrtion of graphs Cor~lkry 1.3. If G is a graph with p vertices, tllzm (1)P x(C, 111 is equd’ to the number of acyclic orientations cbf G. In [ 51, the following question was raise (for a special class of graphs). Let G be a p-vertex graph and let cw) be a labeling of G, i.e., a bijection w : r7(G) + { 1,2, . . . . p}. Define an equivalence relation on the set of all p! labelings w of G by the condition that o w’ if whenever {rcr, U) E X(G), then w(u) < w(v) * O’(U) < ~ti’(u). How many equivalence classes of labelings of G are there? Clearly two labelings o and ~3’ are equivalent if and only if the unique orientations 0 zlnd 0’ compatible with w and o’, respectively, are equal. Moreover, the orientations 0 which arise in this way are precisely the acyclic ones. Hence, by Corollary 1.3, the number of equivalence classes is (1)P XI(G, 1). We conclude this section by iiiscussing the relationship between the . chromatic polynomial of a graph and the order polynomial [4;5;6] of a partially ordered set. If P is a p-element partially ordered set, define the order polynomial a(P, X) (evaluated at the non-negative integer X) to be the number of order-preserving maps u : P -+ { 1,2, . . . . A}. Define the strict’ order polynomial a(P, X) to be the number of strict orderpreserving maps u : P -* { 1,2, . . . . ‘n), i.e., if x < y in P, then a(x) < o(y). In [ 51, it wa:,& shown that fi and fi are pol?rgomials in X related by fi(P, A) z= (1); ,3 W, -4). This is the precise analogue of Theorem 1.2. We shall now ckliy ms analogy. If 0 if;8 an orientation of a graph G, regard 0 as a binary relation > on V(G) defined by u > v if u + v. If 0 is acy Aic, then the transitive and reflexive closure fi of 0 is a partial ordering of t’(G). Moreover, a map 0: V(G) + { 1,2, ‘.., x) is compatible with (3 if and only ii? o is orderpreserving *when considered as a map from a. Hence the number of o compatible with 0 is just a(& A) and we corclude that j&T, A) = c qii, A), u where the sum is over all acyclic orientations 0 of G. In the same way, using Proposition 1.1, we deduce 2. h’numeration of labeled acyclic dkgraphs 175 Hence, Theorem 1.2 follows from the known result a(P, X) = (-l)Pa(P, -h), but we thought a direct proof to be more illuminating. Equation ( 1) strengthens the claim made in [ 4 ] that the strict order polynomial fi is a partially-ordered set analogue of the chromatic polynomial x. 2. Enumeration of labeled acyclic digraphs Corollary I.? , when combined with a result of Read (also obtained by Bender and Goldman), yields an immediate solution to the problem of enumerating labeled acyclic digraphs with n vertices. The same re:ulr was obtained by R.W. Robinson (to be published), who applies it to the unlabeled case. Proposition 2.1. Let f (n) be the number of labeled acyclic digraphs with n vertices. Then 2 f(n) xjU/n! $3 = ( 5 (-l)nxn/n! 2(1;) -’ > . n=O n=O