Related papers: Infinite grids in digraphs

We introduce torsoids, a canonical structure in matching covered graphs, corresponding to the bricks and braces of the graph. This allows a more fine-grained understanding of the structure of finite and infinite directed graphs with respect…

We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every directed graph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a set of at most $f(k)$…

Let $ D $ be a finite digraph, and let $ V_0,\dots,V_{k-1} $ be nonempty subsets of $ V(D) $. The (strong form of) Edmonds' branching theorem states thatthere are pairwise edge-disjoint spanning branchings $ \mathcal{B}_0,\dots,…

An oriented graph is an orientation of a simple graph. In 2009, Keevash, K\"{u}hn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, K\"{u}hn and Osthus showed that…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

Total coloring of a graph is a coloring of its vertices and edges such that adjacent or incident elements receive distinct colors. Total coloring conjecture (stipulating that the total chromatic number of a graph $G$ is at most…

We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all…

Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and…

We prove that for every digraph $F$ and every assignment of pairs of integers $(r_e,q_e)_{e \in A(F)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number at least $N$ contains a subdivision of $F$ in…

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.

Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertices of each hyperedge is partitioned into two parts, a tail…

Let $B$ be a bidirected multigraph with signing $\sigma$, let $X$ be a set of vertices in $B$, and let $k$ be a non-negative integer. For any pair of vertex sets $S,T\subset V(B)$ satisfying $X\cap S = X\cap T$, we denote by $B_{S,T}$ the…

We prove a conjecture of Ohba which says that every graph $G$ on at most $2\chi(G)+1$ vertices satisfies $\chi_\ell(G)=\chi(G)$.

In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

We give a sufficient condition on totally disconnected topological graphs such that their associated topological graph algebras are purely infinite.

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

We prove a conjecture by Aboulker, Charbit and Naserasr by showing that every oriented graph in which the out-neighborhood of every vertex induces a transitive tournament can be partitioned into two acyclic induced subdigraphs. We prove…

We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+o(n)$. This can be seen as a directed graph…