Related papers: An update on the Hirsch conjecture

In 1933, Karol Borsuk asked whether each bounded set in the $n$-dimensional Euclidean space can be divided into $n$+1 parts of smaller diameter. Because it would not make sense otherwise, one usually assumes that he just forgot to require…

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is…

We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$…

We give a new proof of a conjecture of Schinzel on the intersection of a subvariety of codimension at least 2 in a power of the multiplicative group with a torus of dimension 1. The proof rests on a geometric B\'ezout's theorem of P.…

A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…

In this article we prove a conjecture of Bezdek, Brass, and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the…

In 1960, Gr\"{u}nbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently,…

An abstract $n$-polytope $\mathcal{P}$ is a partially-ordered set which captures important properties of a geometric polytope, for any dimension $n$. For even $n \ge 2$, the incidences between elements in the middle two layers of the Hasse…

For $d \ge 4$, the Noether-Lefschetz locus $\mathrm{NL}_d$ parametrizes smooth, degree $d$ surfaces in $\mathbb{P}^3$ with Picard number at least $2$. A conjecture of Harris states that there are only finitely many irreducible components of…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational…

Let $K$ be a number field and $f_1,\ldots,f_s\in K[x_1,\ldots,x_n]$ forms of odd degrees. In 1957, Birch proved that if $n$ is sufficiently large then the forms always have a nontrivial zero in $K^n$. Apart from some small degrees, the…

A graph has \emph{diameter} D if every pair of vertices are connected by a path of at most D edges. The Diameter-D Augmentation problem asks how to add the a number of edges to a graph in order to make the resulting graph have diameter D.…

The construction of the COMBINATORIAL data for a surface with n vertices of maximal genus is a classical problem: The maximal genus g=[(n-3)(n-4)/12] was achieved in the famous ``Map Color Theorem'' by Ringel et al. (1968). We present the…

Let $\mathbb{R}^n$ be the n-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant $1$ such that there is a sphere $|X|<R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly…

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169…

In 2008, Chen and Chv\'atal conjectured that in every finite metric space of $n$ points, there are at least $n$ distinct lines, or the whole set of points is a line. This is a generalization of a classical result in the Euclidean plane. The…

In this paper we study various scribability problems for polytopes. We begin with the classical $k$-scribability problem proposed by Steiner and generalized by Schulte, which asks about the existence of $d$-polytopes that cannot be realized…

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the…