Related papers: Every $d(d+1)$-connected graph is globally rigid i…

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected…

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the…

Among other things, it is shown that for every pair of positive integers $r$, $d$, satisfying $1<r<d\leq 2r$, and every finite simple graph $H,$ there is a connected graph $G$ with diameter $d$, radius $r$, and center $H.$

Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a \emph{finite} subset $A$ in $3$-space so that the vertices of each graph $G$ as above can be mapped into $A$ to…

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carath\'eodory and Tverberg theorems, and their relatives. We conjecture that when the family has at…

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $\alpha(G)=2$ is closely related to Hadwiger's conjecture for graphs…

We investigate the problem of reconstructing a set $P\subseteq \mathbb{R}$ of distinct points, where the only information available about $P$ consists of the distances between some of the pairs of points. More precisely, we examine which…

A $1$-factorization of a graph $G$ is a collection of edge-disjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$-factorization is that $|V(G)|$ is even and $G$ is regular; the converse is…

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick,…

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph…

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…

In this note we prove the conjecture of \cite{HaWiWi} that every bipartite multigraph with integer edge delays admits an edge colouring with $d+1$ colours in the special case where $d=3$. A connection to the Brualdi-Ryser-Stein conjecture…

A $b$-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the maximum…

In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of…

A graph $G$ is embeddable in $\mathbb{R}^d$ if vertices of $G$ can be assigned with points of $\mathbb{R}^d$ in such a way that all pairs of adjacent vertices are at the distance 1. We show that verifying embeddability of a given graph in…

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally…

In this note, we prove that every 4-connected optimal 2-planar graph is Hamiltonian-connected. Furthermore, we show that the 4-connectedness condition is sharp by constructing infinitely many 3-connected optimal 2-planar graphs that are…

Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an…

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erd\H{o}s and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close…