Seminar Series in Mathematics: Algebra 200, 1 7 DIRAC S THEOREM ON CHORDAL GRAPHS Let G be a finite graph. Definition 1. The graph G is called chordal, if each cycle of length has a chord. Theorem 2 (Dirac). The following conditions are equivalent: (a) G is a chordal graph; (b) G is the pure 1-skeleton of a quasi-forest. In this lecture we give an algebraic proof of this theorem as it can be found in the joint paper [12] with Hibi and Zheng. In a first step we characterize a quasi-forest by its Alexander dual. Theorem. Let be a simplicial complex. Then is a quasi-forest if and only if projdimi = 1. The proof of this theorem will be postponed. Let F [n]. We set F c = [n] \ F, and let c = F c : F F ( ). Lemma. Let be a simplicial complex. Then I = I( c ). Proof. The monomial x F is a generator of I, if and only if F [n] is minimal with F, if and only if F [n] is minimal with F c. This is the case if and only if F c F ( ). Thus for the proof of Theorem we have to show: is a quasi-forest projdimi( c ) = 1. Let I = (u 1,...,u m ) be a monomial ideal with minimal set of monomial generators G(I) = {u 1,...,u m }, and let T be the Taylor complex of I. This lecture was held by Jürgen Herzog University of Essen, Germany e-mail: juergen.herzog@uni-essen.de 1
2 DIRAC S THEOREM ON CHORDAL GRAPHS One has with 2 (e i e j ) = u ji e i u i j e j where T 2 2 T1 I 0 u i j = u i gcd(u i,u j ) and u ji = u j gcd(u i,u j ). These ( n 2) relations are called the Taylor relations of I. It is easy to see that there is a minimal multigraded free resolution F of I such that the kernel of F 1 I is generated by a subset of the Taylor relations. Lemma 5. Let = F 1,...,F m. Then the following conditions are equivalent: (a) is a quasi-forest; (b) There exists an (m 1) m submatrix A of the ( m 2) m matrix of Taylor relations of I( c ) = (x F c 1,...,x F c m ) such that I m 1 (A) = I( c ). Here I m 1 (A) is the ideal of the (m 1)-minors of A. Now we are in the position to prove Theorem. By the previous lemma, if is a quasi-forest, then I m 1 (A) = I( c ). By the Hilbert- Burch theorem [1, Theorem 1..17] 0 S m 1 A S m I( c ) 0 (1) is the free resolution of I( c ). Hence projdimi( c ) = 1. Conversely, if projdimi( c ) = 1, then one has a resolution (1) where A is a submatrix of the matrix of Taylor relations of I( c ). Therefore by the previous lemma, is a quasi-forest. We now describe the next steps in the proof of Dirac s theorem. Let G be a finite graph. Definition 6. (a) The complementary graph Ḡ of G is the graph with V (Ḡ) = V (G), and E(Ḡ) = {(i, j): i j and{i, j} E(G)}. (b) A stable subset of G is a subset F [n] such that {i, j} E(G) for all i, j F with i j. We denote by (G) the set of stable subsets of G. Note that (G) is the highest dimensional simplicial complex whose pure 1-skeleton is G.
DIRAC S THEOREM ON CHORDAL GRAPHS G (G) Figure 1: Lemma 7. Let G be a graph, and be the simplicial complex with I = I(Ḡ). then (a) = (G); (b) G = (1); (c) is a quasi-forest G is chordal. Proof. (a) Since (1) = G it follows that I(Ḡ) I (G). Conversely, let F be a minimal nonface of (G). If F > 2, and H F with H = 2, then H is an edge of G, a contradiction. Hence all the minimal nonfaces of (G) are of size 2. This implies that I (G) I(Ḡ). Hence we have I (G) = I(Ḡ), and so = (G). (b) Since = (G) it follows that (1) = G. (c) By a theorem of Fröberg [10], the graph G is chordal if and only if I(Ḡ) = I has a 2-linear resolution. The theorem of Eagon and Reiner (Theorem 20 in [8]) implies then that 2 = reg(i ) = projdimi + 1. Therefore G is chordal if and only if projdimi = 1, which by Theorem is the case if and only if = (G) is a quasi-forest. Lemma 8. A quasi-forest is flag, i.e. I is generated by quadratic monomials. Proof. Choose a leaf ordering F 1...,F m. We use induction on m. Since = F 1,...,F m 1 is a quasi-forest, is flag by induction. Let F k, k < m be a branch of F m. Then consists of all faces G of with G (F m \ F k ) = /0. Suppose H is a minimal nonface of with H. Since H is a nonface, there is p H with p F m. If q F m belongs to H, then {p,q}. Thus there is F j with j m
DIRAC S THEOREM ON CHORDAL GRAPHS such that {p,q} F j. Hence {q} F m F j. Thus q F k. Hence H (F m \ F k ) = /0. This shows that H is a minimal nonface of, a contradiction. We are now in the position to give the proof of Dirac s theorem: Lemma 7, (b) and (c) implies that a chordal graph is the pure 1-skeleton of a quasiforest. This yields the implication (a) (b) in Dirac s theorem. (b) (a): Suppose G is the pure 1-skeleton of a quasi-forest Γ. By Lemma 8, Γ is flag, so that I Γ is generated by monomials x F with F = 2, F Γ. Hence I Γ = I(Ḡ). Then by Lemma 7 (a), Γ = (G), and by (c) G is chordal. At the end of this lecture we want to discuss for a quasi-forest the relation tree of I. By Theorem we have projdimi = 1. Consider any monomial ideal I of projective dimension 1 which is generated by m elements, and assume that the elements of the generating set G(I) have no common factor. Then I is perfect of codimension 2. A subset R of the Taylor relations is called irreducible if R generates the first syzygy module syz 1 (I) of I, but no proper subset of R generates syz 1 (I). It is known (see [2, Corollary 5.2]) that any irreducible subset of the Taylor relations is in fact a minimal system of generators of syz 1 (I). However the choice of an irreducible set R of Taylor relations is in general not unique. For example, let I = (x x 5 x 6,x 1 x 5 x 6,x 1 x 2 x 6,x 1 x 2 x 5 ). Then syz 1 (I) can be represented by the matrix x 1 x 0 0 0 x 2 x 5 0, 0 x 2 0 x 6 or by x 1 x 0 0 0 x 2 x 5 0 0 0 x 5 x 6 or x 1 x 0 0 0 x 2 0 x 6. 0 0 x 5 x 6 Nevertheless for a given choice R of m 1 Taylor relations which generate syz 1 (I) we define a (1-dimensional) tree Ω with {i, j} E(Ω) if u ji e i u i j e j R for i < j. We call Ω the relation tree of R. This relation tree was first considered in [2, Remark 6.]. In the above example the relation tree for the first matrix is given in Figure 2, while for the other matrices it is given in Figure Next we want to describe how the generators u i of I can be computed from the u i j and the relation trees. To this end we introduce for each i = 1,...,m an orientation to make Ω a directed graph which we denote Ω i. We fix some vertex i. Let j be any other vertex of Ω. Since Ω is a tree there is a unique directed walk from i to j. This
DIRAC S THEOREM ON CHORDAL GRAPHS 5 Figure 2: and Figure : defines the orientation of the edges along this walk. The following picture explains this for the first of our relation trees in the above example. By the Hilbert Burch theorem one has u i = ( 1) i det(a i ) for i = 1,...,t, where the matrix A i is obtained from the relation matrix A of I by deleting the ith column of A. Computing det(a i ) by the determinantal expansion formula one sees that u i = u k j, (k, j) where the product is taken over all oriented edges (k, j) of Ω i. In our example I may be viewed as I = I( c ) where the facets of are {{1,2,},{2,,},{,,5},{,,6}}., Figure :
6 DIRAC S THEOREM ON CHORDAL GRAPHS see Figure 5. This is a quasi-forest (in fact, a quasi-tree), as it should be by Theorem. 2 6 1 Figure 5: 5 All possible relation trees Ω of I( c ) can be recovered from the quasi-forest = F 1,...,F m as follows: start with some leaf F i of, and let F j be a branch of F i. Then {i, j} will be an edge of Ω. It follows easily from Dirac s theorem that F ( )\{F i } is again a quasi-forest. Then remove the leaf F i, and continue in the same way with the remaining quasi-forest in order to find the other edges of Ω. Of course, at each step of the procedure there may be different choices. This gives us the different possible relation trees. Geometrically a relation tree is obtained from a given quasi-forest by connecting the barycenters of the leaves and branches according to the above rules. In our example we get Figure 6: References [1] W. Bruns and J. Herzog, Cohen Macaulay rings, Revised Edition, Cambridge University Press, 1996. [2] W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Camb. Phil. Soc., 118 (1995), 25 257. [] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 8 (1961), 71 76.
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