Related papers: A local criterion for Tverberg graphs

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor in $D$, while $D$ is a paired-dominating set of $G$ if $D$ is a~dominating set and the subgraph induced by $D$ contains a perfect matching. A…

In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A…

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C…

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an…

A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks…

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all…

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a…

We prove that there exists an absolute constant $C>0$ such that, for any positive integer $k$, every graph $G$ with minimum degree at least $Ck$ admits a vertex-partition $V(G)=S\cup T$, where both $G[S]$ and $G[T]$ have minimum degree at…

The degree sequence of a graph is the sequence of the degrees of its vertices. If $\pi$ is a degree sequence of a graph $G$, then $G$ is a realization of $\pi$ and $G$ realizes $\pi$. Determining when a sequence of positive integers is…

One of the major results of [N. Robertson and P. D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65--110, 1995], also known as the weak structure theorem, revealed the local structure of graphs…

A new, constructive proof with a small explicit constant is given to the Erd\H{o}s-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at…

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…

Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the…

We say that a digraph is a $(t,\lambda)$-liking digraph if every $t$ vertices have exactly $\lambda$ common out-neighbors. In 1975, Plesn\'{i}k [Graphs with a homogeneity, 1975. {\it Glasnik Mathematicki} 10:9-23] proved that any…

A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Gr\'osz, Methuku and Tompkins in 2020 showed that for any graph $F$, there is an integer $r_0=r_0(F)$, such that for any $r\ge r_0$, any $r$-uniform…

An \emph{independent transversal} (IT) in a graph $G$ with a given vertex partition $P$ is an independent set of vertices of $G$ (i.e. it induces no edges), that consists of one vertex from each part (\emph{block}) of $P$. Over the years,…

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of…

In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…

We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate…