Eran Nevo

Lenses in arrangements of pseudo-circles and their applications

A collection of simple closed Jordan curves in the plane is called a family of <i>pseudo-circles</i> if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudo-circles is said to be an <i>empty lens</i> if the closed Jordan region that it bounds does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of <i>n</i> pseudo-circles with the property that any two curves intersect precisely twice. We use this bound to show that any collection of <i>n</i> <i>x</i>-monotone pseudo-circles can be cut into <i>O</i>(<i>n</i>^8/5) arcs so that any two intersect at most once; this improves a previous bound of <i>O</i>(<i>n</i>^5/3) due to Tamaki and Tokuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced to <i>O</i>(<i>n</i>^3/2(log <i>n</i>)^{<i>O</i>(α(^<i>s</i>(<i>n</i>))}), where α(<i>n</i>) is the inverse Ackermann function, and <i>s</i> is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudo-circles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to <i>O</i>(<i>n</i>^4/3). As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, of pairwise intersecting pseudo-circles, of arbitrary <i>x</i>-monotone pseudo-circles, of parabolas, and of homothetic copies of any fixed simply shaped convex curve. We also obtain a variant of the Gallai--Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound on the number of distinct distances under any well-behaved norm.

The γ-vector of a barycentric subdivision

We prove that the @c-vector of the barycentric subdivision of a simplicial sphere is the f-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex.

Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is claw free with no induced 4-cycle in its complement. For the last family we show that the second power of the edge ideal has a linear resolution.

On γ -Vectors Satisfying the Kruskal–Katona Inequalities

We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement of Gal’s conjecture, which asserts that such γ-vectors are nonnegative.

On r-stacked triangulated manifolds

The notion of r-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the r-stackedness for triangulated homology manifolds and study its basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.

Rigidity and the Lower Bound Theorem for Doubly Cohen–Macaulay Complexes

Abstract We prove that for d≥3, the 1-skeleton of any (d−1)-dimensional doubly Cohen–Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121–125, 1971; Barnette, D. Pac. J. Math. 46:349–354, 1973) hold for every 2-CM complex of dimension ≥2 (see Kalai, G. Invent. Math. 88:125–151, 1987). Moreover, the initial part (g0,g1,g2) of the g-vector of a 2-CM complex (of dimension ≥3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305–1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.

On embeddability and stresses of graphs

Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K5-minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2-minor free graph is generically r-stress free for 1≤r≤4. (This assertion is false for r≥6.) Some further extensions are discussed.

Many triangulated odd-dimensional spheres

It is known that the

Bipartite minors

(2k-1)