Related papers: The Ferrers bound for spanning trees in bipartite …

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this…

Chung and Graham [J. London Math. Soc., 1983] claimed that there exists an $n$-vertex graph $G$ containing all $n$-vertex trees as subgraphs that has at most $\frac{5}{2}n \log_2 n + O(n)$ edges. We identify an error in their proof. This…

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry,…

Let $N\geq 2$ be an integer, a (1, $N$)-periodic graph $G$ is a periodic graph whose vertices can be partitioned into two sets $V_1=\{v\mid\sigma(v)=v\}$ and $V_2=\{v\mid\sigma^i(v)\neq v\ \mbox{for any}\ 1<i<N\}$, where $\sigma$ is an…

Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where $n\geqslant3$. In this paper, explicit expressions for the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ are…

Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. How large does $\alpha$ need to be to guarantee that $G$ contains a connected transversal, that is, a tree on $r$ vertices meeting each…

We prove a conjecture of Shteiner and Shteyner stating that for a bipartite graph $G=(V,E)$, the number of forests in $G$ equals the number of degree sequences arising from its spanning subgraphs. In the process, we provide several…

We establish a new criterion for a bigraph to have a subgraph with prescribed degree conditions. We show that the bigraph $G[X,Y]$ has a spanning subgraph $F$ such that $g(x)\leq deg_F(x) \leq f(x)$ for $x\in X$ and $deg_F(y) \leq f(y)$ for…

For a given $2$-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $B_G(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i$. We determine, for all…

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…

Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have…

Let $G$ be a connected graph with vertex set $V(G)$, and denote by $d_G(u,v)$ the distance from $u$ to $v$ in $G$, for any $u,v \in V(G)$. The average distance of an $n$-vertex connected graph $G$, denoted by $\mu(G)$, is defined to be the…

The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of…

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work…

For a graph $G$ on $n$ vertices, denote by $a(G)$ the number of vertices in the largest induced forest in $G$. The Albertson-Berman conjecture, which has been open since 1979, states that $a(G) \geq \frac{n}{2}$ for every simple planar…

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…

A bipartite graph $H$ is said to have Sidorenko's property if the probability that the uniform random mapping from $V(H)$ to the vertex set of any graph $G$ is a homomorphism is at least the product over all edges in $H$ of the probability…

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2…

A tree is said to be even if for every pair of distinct leaves, the length of the unique path between them is even. In this paper we discuss the problem of determining whether an input graph has a spanning even tree. Hofmann and Walsh…

Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the…