Related papers: Enlarging vertex-flames in countable digraphs
Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to hypergraphs. We introduce a variant of…
A digraph is connected-homogeneous if every isomorphism between two finite connected induced subdigraphs extends to an automorphism of the whole digraph. In this paper, we completely classify the countable connected-homogeneous digraphs.
The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…
We show that every graph admits a canonical tree-like decomposition into its $k$-edge-connected pieces for all $k\in\mathbb{N}\cup\{\infty\}$ simultaneously.
Lov\'{a}sz et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident…
It has been conjectured that if a finite graph has a vertex coloring such that the union of any two color classes induces a connected graph, then for every set $T$ of vertices containing exactly one member from each color class there exists…
A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…
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…
A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on the same vertex set as $G$, if there exists edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound…
Halin proved that every graph with an end $\omega$ containing infinitely many pairwise disjoint rays admits a subdivision of the infinite quarter-grid as a subgraph where all rays from that subgraph belong to $\omega$. We will prove a…
We consider two decomposition problems in directed graphs. We say that a digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its connected components contains at most $k$ arcs. For the first problem, a directed linear…
While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits…
A procedure called \textit{graph burning} was introduced to facilitate the modelling of spread of an alarm, a social contagion, or a social influence or emotion on graphs and networks. Graph burning runs on discrete time-steps (or rounds).…
Thomas proved that every undirected graph admits a linked tree decomposition of width equal to its treewidth. In this paper, we generalize Thomas's theorem to digraphs. We prove that every digraph G admits a linked directed path…
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 $D$…
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…
Let $G$ be an infinite graph such that each tree in the wired uniform spanning forest on $G$ has one end almost surely. On such graphs $G$, we give a family of continuous, measure preserving, almost one-to-one mappings from the wired…
For finite q, we classify the countable, descendant-homogeneous digraphs in which the descendant set of any vertex is a q-valent tree. We also give conditions on a rooted digraph G which allow us to construct a countable…
We introduce the concept of a family of finite directed graphs (\emph{positive integer order,} $f(x) = mx + c; x,m \in \Bbb N$ and $c \in \Bbb N_0)$ which are directed graphs derived from an infinite directed graph called the $f(x)$-root…
A graph with vertex set V and edge set E is called a (d,c)-expander if the maximum degree of a vertex is d and, for every subset W of V that has cardinality at most |V|/2, the number of edges between vertices in W and vertices outside of W…