Related papers: Spanning hypertrees, vertex tours and meanders
We show that a one-ended, locally finite, measurable graph on a standard probability space admits a measurable one-ended spanning subtree if and only if it is measure-hyperfinite. This answers a question posed by Bowen, Poulin, and Zomback…
In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, $R-$graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi \emph{et al.}, 2017), and…
A hypergraph $\mathcal{H}=(V,\mathcal{E})$ is a hypertree if it admits a tree $T$ with vertex set $V$ such that every edge of $\mathcal{H}$ induces a subtree of $T$. A tree like that is called a host tree. Several characterizations and…
We say two spanning trees of a graph are completely independent if their edge sets are disjoint, and for each pair of vertices, the paths between them in each spanning tree do not have any other vertex in common. Pai and Chang constructed…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the…
In this paper two new graph operations are introduced, and with them the S-trees are studied in depth. This allows to find \(\{-1,0,1\}\)-basis for all the fundamental subspaces of the adjacency matrix of any tree, and to understand in…
In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids.…
In this paper, we define a class of auxiliary graphs associated with simple undirected graphs. This class of auxiliary graphs is based on the set of spanning trees of the original graph and the edges constituting those spanning trees. A…
Cayley's formula states that there are $n^{n-2}$ spanning trees in the complete graph on $n$ vertices; it has been proved in more than a dozen different ways over its 150 year history. The complete graphs are a special case of threshold…
Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a…
We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…
Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is…
Hypergraphs require higher-dimensional representations, which makes it more difficult to compute and interpret their spectral properties. This survey article uses the framework of hypermatrices to give an in-depth overview of the spectral…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…
Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
We show that if a graph admits a packing and a covering both consisting of $\lambda$ many spanning trees, where $\lambda$ is some infinite cardinal, then the graph also admits a decomposition into $\lambda$ many spanning trees. For finite…
In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose…
We study that over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees. Trees minimizing (resp. maximizing) the total number of subtrees usually maximize (resp. minimize) the…