We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*max{D*logn,n^ε} for some constant ε>0, then with high probability the random graph G(n,p) contains a copy of T. The obtained bound on the edge probability is shown to be essentially tight for D=n^{Θ(1)}.
We classify connected spanning convex subgraphs of the square cycles. We then show that every spanning tree of $C_n^2$ is contained in a unique nontrivial connected spanning convex subgraph of $C_n^2$. As a result, we obtain a purely combinatorial derivation of the formula for the number of spanning trees of the square cycles.
Halin conjectured 20 years ago that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. We prove Halin's conjecture. This implies a forbidden minor characterisation for the property of having a normal spanning tree.
Schmidt characterised the class of rayless graphs by an ordinal rank function, which makes it possible to prove statements about rayless graphs by transfinite induction. Halin asked whether Schmidt's rank function can be generalised to characterise other important classes of graphs. We answer Halin's question in the affirmative. Another largely open problem raised by Halin asks for a characterisation of the class of graphs with an end-faithful spanning tree. A well-studied subclass is formed by the graphs with a normal spanning tree. We determine a larger subclass, the class of normally tracea…
For integer $k\geq2,$ a spanning $k$-ended-tree is a spanning tree with at most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra [Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237], we provide tight spectral conditions to guarantee the existence of a spanning $k$-ended-tree in a connected graph of order $n$ with extremal graphs being characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we also present tight spectral conditions for the existence of a spann…
A 2-tree is a graph that can be formed by starting with a triangle and iterating the operation of making a new vertex adjacent to two adjacent vertices of the existing graph. Leizhen Cai asked in 1995 whether every maximal planar graph contains a spanning 2-tree. We answer this question in the negative by constructing an infinite class of maximal planar graphs that have no spanning 2-tree.
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 paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rflo…
We present two short proofs for Diestel's criterion that a connected graph has a normal spanning tree provided it contains no subdivision of a countable clique in which every edge has been replaced by uncountably many parallel edges.
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number of spanning trees in specific type of graphs such as product of graphs or Cartesian product of two graphs.
We consider all spanning trees of a complete simple graph $Γ$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $τ(F)$, doesn't depend on the structure of $F$ and is completely determined by the number of vertices $q_i \, (i=1, ..., m)$ in each connected component of $F$. Specifically, $τ(F) = q_1 q_2 \cdots q_m n^{m-2}$.
In this paper, we derive closed-form formulas for Kirchhoff index and Wiener index of linear pentagonal cylinder graph and linear pentagonal Mobius chain graph. We also obtain explicit formulas for finding total number of spanning trees for both the graphs.
It is well-known that the number of spanning trees, denoted by $τ(G)$, in a connected multi-graph $G$ can be calculated by the Matrix-Tree theorem and Tutte's deletion-contraction theorem. In this short note, we find an alternate method to compute $τ(G)$ by degrees of vertices.
In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let $(X,Y)$ be the bipartition of the complete bipartite graph $K_{m,n}$ with $|X|=m$ and $|Y|=n$. We prove that for any given spanning forest $F$ of $K_{m,n}$ with components $T_1,T_2,\ldots,T_k$, the number of spanning trees in $K_{m,n}$ which contain all edges in $F$ is equal to $$ \frac 1{mn}\left(\prod_{i=1}^k (m_in+n_im)\right) \left (1-\sum_{i=1}^{k}\frac{m_in_i}{m_in+n_im}\right ), $$ where $m_i=|V(T_i)\cap X|$ and $n_i=|V(T_i…
