Related papers: Spanning Trees in 2-trees

Let $G_1$ and $G_2$ be simple graphs and let $n_1 = |V(G_1)|$, $m_1 = |E(G_1)|$, $n_2 = |V(G_2)|$ and $m_2 = |E(G_2)|.$ In this paper we derive sharp upper and lower bounds for the number of spanning trees $\tau$ in the Cartesian product…

Let $G$ be a graph with a spanning subgraph $F$, let $m$ be a positive integer, and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S\subseteq V(G)$, $$\Omega_m(G\setminus S)\le \sum_{v\in…

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in…

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is…

For a vertex $v$ of a graph $G$, a spanning tree $T$ of $G$ is distance-preserving from $v$ if, for any vertex $w$, the distance from $v$ to $w$ on $T$ is the same as the distance from $v$ to $w$ on $G$. If two vertices $u$ and $v$ are…

We prove that the spanning trees of any outerplanar triangulation $G$ can be listed so that any two consecutive spanning trees differ in an exchange of two edges that share an end vertex. For outerplanar graphs $G$ with faces of arbitrary…

Let $T$ be an oriented tree on $n$ vertices with maximum degree at most $e^{o(\sqrt{\log n})}$. If $G$ is a digraph on $n$ vertices with minimum semidegree $\delta^0(G)\geq(\frac12+o(1))n$, then $G$ contains $T$ as a spanning tree, as…

We present a determinantal formula for the number of spanning trees of a complete multipartite graph containing a given spanning forest $F$. Our approach relies on the Generalized Matrix Determinant Lemma and Jacobi's formula for the…

We prove Ehrenborg's conjecture that every connected bipartite graph $G$ with parts of size $m$ and $n$ has at most $\frac{1}{mn}\prod_{v\in V(G)} \operatorname{deg}(v)$ spanning trees, and that equality holds if and only if $G$ is a…

For a graph $G$, let $\tau(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $\tau(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for…

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ such that any nontrivial path whose endpoints have the same label contains a larger label. The least $k$ for which $G$ has a $k$-ranking is the ranking number…

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…

Let $G=(V,E)$ be a loopless graph and $\mathcal{T}(G)$ be the set of all spanning trees of $G$. Let $L(G)$ be the line graph of the graph $G$ and $t(L(G))$ be the number of spanning trees of $L(G)$. Then, by using techniques from electrical…

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. For two distinct vertices $u,v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v.$ For a…

The circumference of a graph $G$ is the length of a longest cycle in $G$, or $+\infty$ if $G$ has no cycle. Birmel\'e (2003) showed that the treewidth of a graph $G$ is at most its circumference minus $1$. We strengthen this result for…

Chung and Graham considered the problem of minimizing the number of edges in an $n$-vertex graph containing all $n$-vertex trees as a subgraph. They showed that such a graph has at least $\frac{1}{2}n \log{n}$ edges. In this note, we…

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…

A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…

We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs…