We describe Bott towers as sequences of toric manifolds M k , and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's analysis of the Adams Spectral Sequence [2]. By way of application we investigate stably complex structures on M k , identifying those which arise from omniorientations and those which are almost complex. We conclude with observations on the rôle of Bott towers in complex cobordism theory.

A vertex coloring of a simplicial complex Δ is called a linear coloring if it satisfies the property that for every pair of facets (F1, F2) of Δ, there exists no pair of vertices (v1, v2) with the same color such that v1 ∈ F1 {set minus} F2 and v2 ∈ F2 {set minus} F1. The linear chromatic numberlchr (Δ) of Δ is defined as the minimum integer k such that Δ has a linear coloring with k colors. We show that if Δ is a simplicial complex with lchr (Δ) = k, then it has a subcomplex Δ′ with k vertices such that Δ is simple homotopy equivalent to Δ′. As a corollary, we obtain that lchr (Δ) ≥ Homdim (Δ) + 2. We also show in the case of linearly colored simplicial complexes, the usual assignment of a simplicial complex to a multicomplex has an inverse. Finally, we show that the chromatic number of a simple graph is bounded from above by the linear chromatic number of its neighborhood complex. © 2007 Elsevier Inc. All rights reserved

We call a vertex $x$ of a graph $G=(V,E)$ a codominated vertex if $N_G[y]\subseteq N_G[x]$ for some vertex $y\in V\backslash \{x\}$, and a graph $G$ is called codismantlable if either it is an edgeless graph or it contains a codominated vertex $x$ such that $G-x$ is codismantlable. We show that $(C_4,C_5)$-free vertex-decomposable graphs are codismantlable, and prove that if $G$ is a $(C_4,C_5,C_7)$-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for $G$ are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity $reg(G)$ of such graphs, and show that $reg(G)=im(G)$ whenever $G$ is a $(C_4,C_5)$-free vertex-decomposable graph, where $im(G)$ is the induced matching number of $G$. Furthermore, we prove that $H$ must be a codismantlable graph if $im(H)=reg(H)=m(H)$, where $m(H)$ is the matching number of $H$. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family $H_n$ ($n\geq 4$) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and $reg(H_n)=im(H_n)$ if $n\geq 6$. This answers a recent question of Mahmoudi et al.

Abstract. We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent bundle of the kth stage of the tower, considered as a complex manifold, which splits into a sum of complex line bundles. Applying Danilov-Jurkiewicz theorem, we compute the cohomology ring of any kth stage, and by way of construction, we provide all the monomial identities defining the related affine toric varieties.

We present new combinatorial insights into the calculation of (Castelnuovo-Mumford) regularity of graphs. We first show that the regularity of any graph can be reformulated as a generalized induced matching problem. On that direction, we introduce the notion of a prime graph by calling a connected graph G as a prime graph over a field k, if reg k pG´xq ă reg k pGq for any vertex x P V pGq. We exhibit some structural properties of prime graphs that enables us to compute the regularity in specific hereditary graph classes. In particular, we prove that regpGq ď ∆pGq impGq holds for any graph G, and in the case of claw-free graphs, we verify that this bound can be strengthened by showing that regpGq ď 2 impGq, where impGq is the induced matching number of G. By analysing the effect of Lozin transformations on graphs, we narrow the search of prime graphs into bipartite graphs having sufficiently large girth with maximum degree at most three, and show that the regularity of bipartite graphs G with such constraints is bounded above by 2 impGq`1. Moreover, we prove that any non-trivial Lozin operation preserves the primeness of a graph that enables us to generate new prime graphs from the existing ones.We introduce a new graph invariant, the virtual induced matching number vimpGq satisfying impGq ď vimpGq ď regpGq for any graph G that results from the effect of edgecontractions and vertex-expansions both on graphs and the independence complexes of graphs to the regularity. In particular, we verify the equality regpGq " vimpGq for a graph class containing all Cohen-Macaulay graphs of girth at least five.Finally, we prove that there exist graphs satisfying regpGq " n and impGq " k for any two integers n ě k ě 1. The proof is based on a result of Januszkiewicz and Swiatkowski [18] accompanied with Lozin operations. We provide an upper bound on the regularity of any 2K 2 -free graph G in terms of the maximum privacy degree of G. In addition, if G is a prime 2K 2 -free graph, we show that regpGq ď δpGq`3 2 .In particular, Theorem 2.6 implies that IndpGq » IndpG´eq whenever the edge e is isolating. This brings the use of an operation, adding or removing an edge, on a graph without altering its homotopy type. We will follow [1] to write Addpx, y; wq (respectively Delpx, y; wq) to indicate that we add the edge e " xy to (resp. remove the edge e " xy from) the graph G, where w is the corresponding isolated vertex.Remark 2.7. In order to simplify the notation, we note that when we mention the homology, homotopy or a suspension of a graph, we mean that of its independence complex, so whenever it is appropriate, we drop Indp´q from our notation. Prime graphs and Prime FactorizationsAs we have already mentioned in Section 1, the notion of primeness brings a new strategy for the calculation of the regularity. Even if we express its definition in Section 1, there seems no harm for restating it in its greatest generality.Corollary 5.14. Let z be a non-isolated vertex of a graph G such that rA z , B z s is a t-pairing in G´N G rzs, then ...