Related papers: Induced graphs of uniform spanning forests

A spanning tree T in a finite planar connected graph G determines a dual spanning tree T* in the dual graph G such that T and T* do not intersect. We show that it is not always possible to find T in G, such that the diameters of T and T*…

An $({\cal F},{\cal F}_d)$-partition of a graph is a vertex-partition into two sets $F$ and $F_d$ such that the graph induced by $F$ is a forest and the one induced by $F_d$ is a forest with maximum degree at most $d$. We prove that every…

We study the existence and construction of sparse supports for hypergraphs derived from subgraphs of a graph $G$. For a hypergraph $(X,\mathcal{H})$, a support $Q$ is a graph on $X$ s.t. $Q[H]$, the graph induced on vertices in $H$ is…

Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the…

We study a random graph $G$ with given degree sequence $\boldsymbol{d}$, with the aim of characterising the degree sequence of the subgraph induced on a given set $S$ of vertices. For suitable $\boldsymbol{d}$ and $S$, we show that the…

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and…

We present and study the following conjecture: for an integer $t\geq 4$ and a graph $H$, every even-hole-free graph of large enough treewidth has an induced subgraph isomorphic to either $K_t$ or $H$, if (and only if) $H$ is a $K_4$-free…

We introduce a graph partitioning problem motivated by computational topology and propose two algorithms that produce approximate solutions. Specifically, given a weighted, undirected graph $G$ and a positive integer $k$, we desire to find…

Let $\mathcal{F}$ be a set of connected graphs, and let $G$ be a graph. We say that $G$ is \emph{$\mathcal{F}$-free} if it does not contain $F$ as an induced subgraph for all $F\in\mathcal{F}$, and we call $\mathcal{F}$ a forbidden pair if…

A graph $G$ is $H$-saturated for a graph $H$, if $G$ does not contain a copy of $H$ but adding any new edge to $G$ results in such a copy. An $H$-saturated graph on a given number of vertices always exists and the properties of such graphs,…

We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every…

An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $\rho(G)$ of $G$ is the minimum number of paths in an induced path…

We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition…

A graph $G=(V,E)$ is said to be odd (or even, resp.) if $d_G(v)$ is odd (or even, resp.) for any $v\in V$. Trivially, the order of an odd graph must be even. In this paper, we show that every 4-edge connected graph of even order has a…

We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an…

Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF…

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete…

Given a graph $H$, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to $H$, if and only if $H$ is a forest.

Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of…