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The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…

A topology $\tau$ on a set $X$ is called maximal connected if it is connected, but no strictly finer topology $\tau^* > \tau$ is connected. We consider a construction of so-called tree sums of topological spaces, and we show how this…

Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a…

In this paper we study the subset sum problem with real numbers. Starting from the given problem, we formulate a quadratic maximization problem over a polytope, P, which is eventually written as a distance maximization to a fixed point over…

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance from $v$ to any other vertex. The vertices whose eccentricity are equal to the diameter (the maximum eccentricity) of $G$ are called peripheral vertices. In trees the…

This paper considers the conjecture by Gr\"unbaum that every planar 3-connected graph has a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 3. Here, the co-tree of $T$ is the spanning tree of the dual…

A metric space $\mathcal{T}$ is a \emph{real tree} if for any pair of points $x, y \in \mathcal{T}$ all topological embeddings $\sigma$ of the segment $[0,1]$ into $\mathcal{T}$, such that $\sigma (0)=x$ and $\sigma (1)=y$, have the same…

THIS IS A CORRECTED VERSION INCLUDING AN APPENDED CORRIGENDUM. Best match graphs arise naturally as the first processing intermediate in algorithms for orthology detection. Let $T$ be a phylogenetic (gene) tree $T$ and $\sigma$ an…

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag…

Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The…

Recently, Adiprasito et al. have initiated the study of the so-called no-dimensional Tverberg problem. This problem can be informally stated as follows: Given $n\geq k$, partition an $n$-point set in Euclidean space into $k$ parts such that…

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…

Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most…

A straight-line drawing $\Gamma$ of a graph $G=(V,E)$ is a drawing of $G$ in the Euclidean plane, where every vertex in $G$ is mapped to a distinct point, and every edge in $G$ is mapped to a straight line segment between their endpoints. A…

The general sum-connectivity index of a graph $G$ is defined as $\chi_\alpha(G)=\sum\limits_{uv\in E(G)} {(d(u)+d(v))^{\alpha}}$, where $d(v)$ denotes the degree of the vertex $v$ in $G$ and $\alpha$ is a real number. In this paper it is…

A graph G=(V,E) is called a unit-distance graph in the plane if there is an injective embedding of V in the plane such that every pair of adjacent vertices are at unit distance apart. If additionally the corresponding edges are non-crossing…

Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have…

The contact graph of a packing of translates of a convex body in Euclidean $d$-space $\mathbb E^d$ is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch…

The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph.…

Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The Sombor index of the graph $G$ is a degree-based topological index, defined as $$SO(G)=\sum_{uv \in E}\sqrt{d(u)^2+d(v)^2},$$ in which $d(x)$ is the degree of the…