Kruskal-Katona type theorems for clique complexes arising from chordal and strongly chordal graphs

A forest is the clique complex of a strongly chordal graph and a quasiforest is the clique complex of a chordal graph. Kruskal–Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. Introduction Recently, in commutative algebra, the forest ([5]) and the quasi-forest ([17] and [9]) have been extensively studied. Each of these concepts is, however, well known in combinatorics ([14]). In fact, a forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. (A chordal graph is a finite graph for which every cycle of length > 3 has a chord. A strongly chordal graph is a chordal graph for which every cycle of even length ≥ 6 has a chord that joins two vertices of the cycle with an odd distance > 1 in the cycle.) Besides the celebrated g-conjecture for spheres ([16, pp. 75–76]), one of the most important open problems in the study of f -vectors of simplicial complexes is the classification of f -vectors of flag complexes. (A flag complex is the clique complex of a finite graph.) Works in the reserach topic include [3], [6], [7], [8] and [15]. On the other hand, the study of f -vectors of clique complexes of chordal graphs was done in [4], [12] and [13]. The purpose of the present paper is to give a Kruskal–Katona type theorem for forests and quasi-forests (Theorem 1.1) as well as a Kruskal–Katona type theorem for pure forests and pure quasi-forests (Theorem 1.2). These theorems will be proved in Section 2. We then show in Section 3 that the f -vector of a pure quasi-forest is unimodal. 1. Kruskal–Katona type theorems Let [n] = {1, . . . , n} be the vertex set and ∆ a simplicial complex on [n]. Thus ∆ is a collection of subsets of [n] with the properties that (i) {i} ∈ ∆ for each i ∈ [n] and (ii) if F ∈ ∆ and G ⊂ F , then G ∈ ∆. Each element F ∈ ∆ is a face of ∆. Let d = max{|F | : F ∈ ∆}, where |F | is the cardinality of F . Then dim∆, the dimension of ∆, is d− 1. A facet is a maximal face of ∆ under inclusion. We write F(∆) for the set of facets of ∆. A simplicial complex is called pure if all facets have the same cardinality. Let fi denote the number of faces F with |F | = i + 1. The vector f(∆) = (f0, f1, . . . , fd−1) is called the f -vector of ∆. In particular f0 = n. If 1 {Fi1 , . . . , Fiq} is a subset of F(∆), then we write 〈Fi1 , . . . , Fiq〉 for the subcomplex of ∆ whose faces are those faces F of ∆ with F ⊂ Fij for some 1 ≤ j ≤ q. A facet F of a simplicial complex ∆ is called a leaf if there is a facet G 6= F of ∆, called a branch of F , such that H ⋂ F ⊂ G ⋂ F for all facets H of ∆ with H 6= F . A quasi-forest is a simplicial complex ∆ which enjoys an ordering F1, F2, . . . , Fs of the facets of ∆, called a leaf order, such that for each 1 < j ≤ s the facet Fj is a leaf of the subcomplex 〈F1, . . . , Fj−1, Fj〉 of ∆. A quasi-tree is a quasi-forest which is connected. A forest is a simplicial complex ∆ which enjoys the property that for every subset {Fi1 , . . . , Fiq} of F(∆) the subcomplex 〈Fi1, . . . , Fiq〉 of ∆ has a leaf. A tree is a forest which is connected. We now come to Kruskal–Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests. Theorem 1.1. Given a finite sequence (f0, f1, . . . , fd−1) of integers with each fi > 0, the following conditions are equivalent: (i) there is a quasi-forest ∆ of dimension d− 1 with f(∆) = (f0, f1, . . . , fd−1); (ii) there is a forest ∆ of dimension d− 1 with f(∆) = (f0, f1, . . . , fd−1); (iii) the sequence (c1, . . . , cd) defined by the formula d ∑ i=0 fi−1(x− 1) i = d ∑ i=0 cix , (1) where f−1 = 1, satisfies ∑d i=k ci > 0 for each 1 ≤ k ≤ d. (iv) the sequence (b1, . . . , bd) defined by the formula d ∑ i=1 fi−1(x− 1) i−1 = d ∑ i=1 bix i−1 (2) is positive, i.e., bi > 0 for 1 ≤ i ≤ d. Theorem 1.2. Given a finite sequence (f0, f1, . . . , fd−1) of integers with each fi > 0, the following conditions are equivalent: (i) there is a pure quasi-forest ∆ of dimension d−1 with f(∆) = (f0, f1, . . . , fd−1); (ii) there is a pure forest ∆ of dimension d− 1 with f(∆) = (f0, f1, . . . , fd−1); (iii) the sequence (c1, . . . , cd) defined by (1) satisfies ∑d i=k ci > 0 for each 1 ≤ k ≤ d and ci ≤ 0 for each 1 ≤ i < d. (iv) the sequence (b1, . . . , bd) defined by the formula (2) satisfies 0 < b1 ≤ b2 ≤ · · · ≤ bd. In Section 2, after preparing Lemmata 2.1, 2.2 and 2.3, we will prove both of Theorems 1.1 and 1.2 simultaneously. 2. f-vectors of forests and quasi-forests We begin with Lemma 2.1. Let ∆ be a quasi-forest on [n] with s + 1 facets and Fs+1, Fs, . . . , F1 its leaf order. For each 1 ≤ j ≤ s we write Gj for a branch of the leaf Fj in the subcomplex 〈Fs+1, Fs, . . . , Fj〉 of ∆. Let δj = |Fj | and ej = |Fj ⋂ Gj |. Let dim∆ = d− 1 and let f(∆) = (f0, f1, . . . , fd−1) be the f -vector of ∆. 2