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Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.
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Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length $n$, the classical associahedra, is $2n-6$ for a large enough $n$. We give a tight bound of $Theta(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is $Theta (nlog n)$.
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $mathbb{F}_q^2$ over a finite field $mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $mathbb{F}_q$. It turns out that integral point sets over $mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $mathbb{F}_q$ for $qle 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.
In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on the diameters of convex polyhedra. Todd and Sukegawa-Kitahara proved tail-quasipolynomial bounds on the diameters of polyhedra. These tail bounds apply when the number of facets is greater than a certain function of the dimension. We prove tail-quasipolynomial bounds on the diameters of polytopes and normal simplicial complexes. We also prove tail-polynomial upper bounds on the diameters of polyhedra.
Given $n$ points in the plane, a emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossi ng) covering path consisting of $n/2 +O(n/log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-eps)n$ straight line segments suffice for a small constant $eps>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing emph{covering trees} is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $Omega(n log{n})$ time in the worst case.
Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,vin X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with $n$ vertices is also provided.
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