Adrian Riskin
In answer to a question of Eggleton, we prove that the complete multigraph on 5 vertices with edge multiplicity 6, namely $K_{5}^{(6)}$, has a decomposition into 5 copies of the family of trees of order 5 and that $K_{7}^{(22)}$ has a decomposition into 7 copies of the family of trees of order 7. We prove something similar for $K_{2n+1}$ for $n \le 13$.
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We give an Ore-Type condition sufficient for a graph G to have a spanning tree with a specified degree sequence.
We find the threshold for the existence of a collection of edge disjoint copies of $K_r$ that form a cyclic structure and span all vertices of $G_{n,p}$. We use a recent result of Riordan to give a two line proof of the main result.
Let $G$ be a simple graph with $n\geq4$ vertices and $d(x)+d(y)\geq n+k$ for each edge $xy\in E(G)$. In this work we prove that $G$ either contains a spanning closed trail containing any given edge set $X$ if $|X|\leq k$, or $G$ is a well characterized graph. As a corollary, we show that line graphs of such graphs are $k$-hamiltonian.
Let $T$ be a tree, a vertex of degree one is called a leaf. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this note, we give a sharp sufficient condition to show that a $K_{1,t}-$free graph has a spanning tree whose stem has a few leaves. By applying the main result, we give improvements of previous related results.
A note on universal graphs for spanning trees
本文针对含所有n顶点树为子图的n顶点通用图的最小边数问题，改进了Chung和Graham给出的1/2 n log n边数下界，将其提升至n log n。
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
The \emph{spanning tree packing number} of a graph $G$ is the maximum number of edge-disjoint spanning trees contained in $G$. Let $k\geq 1$ be a fixed integer. Palmer and Spencer proved that in almost every random graph process, the hitting time for having $k$ edge-disjoint spanning trees equals the hitting time for having minimum degree $k$. In this paper, we prove that for any $p$ such that $(\log n+ω(1))/n\leq p\leq (1.1\log n)/n$, almost surely the random graph $G(n,p)$ satisfies that the spanning tree packing number is equal to the minimum degree. Note that this bound for $p$ will allow …
A short note on spanning even trees
本文证明每个奇数度 r-正则非二分连通图都存在生成偶树，补上 Jackson–Yoshimoto 猜想在无 2-因子保证下的最后未决情形。
For a spanning tree T of a connected graph G and for a labelling ϕ: E(T) \rightarrow {+, -}, ϕis called an alternating sign on a spanning tree T of a graph G if for any cotree edge e \in E(G)-E(T), the unique path in T joining both end vertices of e has alternating signs. In the present note, we prove that any graph has a spanning tree T and an alternating sign on T.
This note derives the characteristic polynomial of a graph that represents nonjump moves in a generalized game of checkers. The number of spanning trees is also determined.
Consider a (not necessarily near-critical) random graph running in continuous time. A recent breadth-first-walk construction is extended in order to account for the surplus edge data in addition to the spanning edge data. Two different graph representations of the multiplicative coalescent, with different advantages and drawbacks, are discussed in detail. A canonical multi-graph of Bhamidi, Budhiraja and Wang (2014) naturally emerges. The presented framework should facilitate understanding of scaling limits with surplus edges for near-critical random graphs in the domain of attraction of gener…
We consider spanning trees of $n$ points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive $2n-Ω(1)$ upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a $2n-Ω(\log n)$ bound. We improve their result with a $2n-Ω(\sqrt{n})$ upper bound, and we strengthen and shorten the proofs of several of their resul…
The bipartite analogue of a classical spanning tree enumeration formula, Boolean functions, and their applications to counting odd spanning trees
本文通过经典生成树计数公式和布尔函数，给出了计算完全图中奇生成树的简单证明，并将其推广到完全二分图中。
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.
Let $G$ be a cubic graph which has a decomposition into a spanning tree $T$ and a $2$-regular subgraph $C$, i.e. $E(T) \cup E(C) = E(G)$ and $E(T) \cap E(C) = \emptyset$. We provide an answer to the following question: which lengths can the cycles of $C$ have if $G$ is a snark? Note that $T$ is a hist (i.e. a spanning tree without a vertex of degree two) and that every cubic graph with a hist has the above decomposition.
On stability of weighted spanning tree degree enumerators
本文将生成树度枚举器的实稳定性结果从连通图推广到加权图，并据此定义了加权距离遗传图类。
We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some $k$, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with $t$ triples spanning $\mathcal{O}(\sqrt{t})$ vertices, which is the best possible up to the implied constant. We confirm that for all $t$ we can find a collection of $t$ triples spanning at most $t+3$ vertices, resolving the Brown--Erd\H os--Sós conjec…
On the number of distinct spanning trees in pseudorandom graphs
研究伪随机图中生成树数量，证明在特定条件下生成树数量下限为(α-ε)^n。
In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every $K_n$ properly edge-coloured by $n-1$ colours has $n/2$ edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-coloured $K_n$. Improving the previous best known bound, we show that every properly edge-coloured $K_n$ contains $Ω(n)$ pairwise edge-disjoint rainbow spanning trees. Independently, Pokrovskiy and…
Note on near-spanning balanced antidirected trees missing from regular tournaments
本文构造了满足半度条件但无法嵌入高度对称锦标赛的近生成平衡反定向树，给出了Stein猜想在均匀次线性度解释下的反例，并确定了最优目标阶及最小宿主阶。
Let ${\mathcal U}_δ$ be the uniform spanning tree on $δ\mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_δ$ is a connected component of the restriction of ${\mathcal U}_δ$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $δ\to 0$, which resolves an open question raised by Benjamini (1999).
We study the height of a spanning tree $T$ of a graph $G$ obtained by starting with a single vertex of $G$ and repeatedly selecting, uniformly at random, an edge of $G$ with exactly one endpoint in $T$ and adding this edge to $T$.
A short note on the order of the double reduced 2-factor transfer digraph for rectangular grid graphs
本文针对矩形网格图中圈的生成并集枚举问题，给出了当m为奇数时双约化2-因子转移有向图阶的精确组合公式。
The number of trees in distance-hereditary graphs and their friends
本文统一证明并推广了2020年代关于距离遗传图生成树计数的若干结果，通过研究顶点复制操作下度枚举子的行为，并证明了Ehrenborg猜想与其多项式形式的等价性。
Choi et. al (2011) introduced a minimum spanning tree (MST)-based method called CLGrouping, for constructing tree-structured probabilistic graphical models, a statistical framework that is commonly used for inferring phylogenetic trees. While CLGrouping works correctly if there is a unique MST, we observe an indeterminacy in the method in the case that there are multiple MSTs. In this work we remove this indeterminacy by introducing so-called vertex-ranked MSTs. We note that the effectiveness of CLGrouping is inversely related to the number of leaves in the MST. This motivates the problem of f…
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