Related papers: Absorbing angles, Steiner minimal trees, and antip…

We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an…

Assume that we embed the path $P_n$ as a subgraph of a $2$-dimensional grid, namely, $P_k \times P_l$. Given such an embedding, we consider the ordered set of subpaths $L_1, L_2, \ldots , L_m$ which are maximal straight segments in the…

The Steiner Tree problem is a classical problem in combinatorial optimization: the goal is to connect a set $T$ of terminals in a graph $G$ by a tree of minimum size. Karpinski and Zelikovsky (1996) studied the $\delta$-dense version of…

We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending…

In this paper, we introduce the notions of Star-$\sigma\mathcal{K}$ and absolutely Star-$\sigma\mathcal{K}$ spaces which allow us to unify results among several properties in the theory of star selection principles on small spaces. In…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by removing less than $k$ vertices. It is separable if there exists a tree-decomposition of adhesion less than $k$ of $G$ in which…

Given a set $P$ of terminals in the plane and a partition of $P$ into $k$ subsets $P_1, ..., P_k$, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree $T_i$ connecting the terminals in each set $P_i$ ($i=1,...,k$)…

Let $k \geq 2$ be an integer. Kouider and Lonc proved that the vertex set of every graph $G$ with $n \geq n_0(k)$ vertices and minimum degree at least $n/k$ can be covered by $k - 1$ cycles. Our main result states that for every $\alpha >…

Two vertex-labelled polygons are \emph{compatible} if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations---for every face, the clockwise cyclic…

We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$…

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal…

The geometric bottleneck Steiner network problem on a set of vertices $X$ embedded in a normed plane requires one to construct a graph $G$ spanning $X$ and a variable set of $k\geq 0$ additional points, such that the length of the longest…

There is a constant c > 0 such that for every $\epsilon \in (0,1)$ and $n \geq 1/\epsilon^2$, the following holds. Any mapping from the $n$-point star metric into $\ell_1^d$ with bi-Lipschitz distortion $1+\epsilon$ requires dimension $$d…

The famous Erd\H{o}s-S\'os conjecture states that every graph of average degree more than $t-1$ must contain every tree on $t+1$ vertices. In this paper, we study a spectral version of this conjecture. For $n>k$, let $S_{n,k}$ be the join…

In this note, we propose a straightforward method to produce an straight-line embedding of a planar graph where one face of a graph is fixed in the plane as a star-shaped polygon. It is based on minimizing discrete Dirichlet energies,…

A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and…

Klimo\v{s}ov\'a, Piguet, and Rozho\v{n} conjectured that any graph with minimum degree $k/2$ and sufficiently many vertices of degree $k$ should contain all trees with $k$ edges. We prove an asymptotic version of this conjecture for dense…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

We derive the low energy absorption cross section of minimal scalars for a very generic class of geometries, that includes, among others, black holes, strings, $p$-branes, as well as intersecting brane configurations. The scalar field can…

We introduce a uniform method of proof for the following results. For {\em each} of the following conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying that condition: i) Theorem~2.2.4: (extending \cite{Chicoetal})…