Related papers: Large vertex-flames in uncountable digraphs

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph…

We prove that if $M$ is a maximal $k$-edge-colorable subgraph of a multigraph $G$ and if $F = \{v \in V(G) : d_M(v) \leq k-\mu(v)\}$, then $d_F(v) \leq d_M(v)$ for all $v \in F$. (When $G$ is a simple graph, the set $F$ is just the set of…

In graph theory, knowing the number of complete subgraphs with r vertices that a graph g has, limits the number of its complete subgraphs with s vertices, for s > r. A useful upper bound is provided by the Kruskal-Katona theorem, but this…

It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end…

We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded degree. This lemma extends a sequence of results of Lov\'asz, Catlin, Kostochka and Rabern.

The claw number of a graph $G$ is the largest number $v$ such that $K_{1,v}$ is an induced subgraph of $G$. Interval graphs with claw number at most $v$ are cluster graphs when $v = 1$, and are proper interval graphs when $v = 2$. Let…

Extending a result of Christiansen, we prove that every mutli-graph $G=(V,E)$ admits a proper edge colouring $\phi:E\to \{1,2,\dots\}$ which is local, that is, $\phi(e)\le \max\{d(x)+\pi(x),d(y)+\pi(y)\}$ for every edge $e$ with end-points…

Given $0<\alpha\leq\pi$, ${\epsilon}>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$…

The Weisfeiler-Leman (WL) dimension is an established measure for the inherent descriptive complexity of graphs and relational structures. It corresponds to the number of variables that are needed and sufficient to define the object of…

It is well known that determining if a digraph has a kernel is an NP-complete problem. However, Topp proved that when subdividing every arc of a digraph we obtain a digraph with a kernel. In this paper we define the kernel subdivision…

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…

Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…

In this work we continue the investigation launched in [FHR16] of the structural properties of the structural properties of the Layers model, a dependent percolation model. Given an undirected graph $G=(V,E)$ and an integer $k$, let…

For any fixed simple graph $H=(V,E)$ and any fixed $u>0$, we establish the leading order of the exponential rate function for the probability that the number of copies of $H$ in the Erd\H{o}s--R\'enyi graph $G(n,p)$ exceeds its expectation…

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly…

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that…

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…

In this paper we introduce a new notion of convergence of sparse graphs which we call Large Deviations or LD-convergence and which is based on the theory of large deviations. The notion is introduced by "decorating" the nodes of the graph…

Conventional spectral digraph partitioning methods typically symmetrize the adjacency matrix, thereby transforming the directed graph partitioning problem into an undirected one, where bipartitioning is commonly linked to minimizing graph…