纯对. IX. 横截树

A bandwidth theorem for graph transversals

Given a collection $\mathcal{G}=(G_1,\dots, G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $λ: E(H) \rightarrow [h]$ such that $e\in E(G_{λ(e)})$ for each $e\in E(H)$. The conditions on the minimum degree $δ(\mathcal{G})=\min_{i\in[h]}\{ δ(G_i)\}$ for finding a spanning $\mathcal{G}$-transversal isomorphic to a graph $H$ have been actively studied when $H$ is a Hamilton cycle, an $F$-factor, a spanning tree with maximum degree $o(n/\log n)$ and a power of a Hamilton cycle, etc. In th…

长定向彩虹循环和彩虹生成树

Long directed rainbow cycles and rainbow spanning trees

边着色图的一个子图称为彩虹，如果它的所有边都有不同的颜色。彩虹子图的发现问题可以追溯到欧拉关于拉丁方中横截的工作，并从那时起得到了广泛的研究。本文研究了完全边着色图和有向图的彩虹子图的两个相关问题。在第一部分中，我们证明了每一个正常边着色的完全有向图都包含一个长度为$n-O(n^{4/5})$的有向彩虹圈。这是由哈恩的一个老问题激发的，改进了Gyarfas和Sarkozy的一个结果。在第二部分中，我们表明，任何树$T$上的$n$顶点与最大程度的$Δ_T\leq βn/\log n$有一个彩虹嵌入到一个适当的边缘着色$K_n$，只要每一种颜色出现在最多的$αn$时间和$α, β$是足够小的常数。

Pure pairs. IV. Trees in bipartite graphs

In this paper we investigate the bipartite analogue of the strong Erdos-Hajnal property. We prove that for every forest $H$ and every $τ>0$ there exists $ε>0$, such that if $G$ has a bipartition $(A,B)$ and does not contain $H$ as an induced subgraph, and has at most $(1-τ)|A|\cdot|B|$ edges, then there is a stable set in $G$ that contains at least $ε|V_i|$ vertices of $V_i$, for $i=1,2$. No graphs $H$ except forests have this property.

Pure pairs. VI. Excluding an ordered tree

A pure pair in a graph $G$ is a pair $(Z_1,Z_2)$ of disjoint sets of vertices such that either every vertex in $Z_1$ is adjacent to every vertex in $Z_2$, or there are no edges between $Z_1$ and $Z_2$. With Maria Chudnovsky, we recently proved that, for every forest $F$, every graph $G$ with at least two vertices that does not contain $F$ or its complement as an induced subgraph has a pure pair $(Z_1,Z_2)$ with $|Z_1|,|Z_2|$ linear in $|G|$. Here we investigate what we can say about pure pairs in an {\em ordered} graph $G$, when we exclude an ordered forest $F$ and its complement as induced su…

Pure pairs. I. Trees and linear anticomplete pairs

The Erdos-Hajnal Conjecture asserts that for every graph H there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. In this paper, we prove a conjecture of Liebenau and Pilipczuk, that for every forest H there exists c > 0, such that every graph G contains either an induced copy of H, or a vertex of degree at least c|G|, or two disjoint sets of at least c|G| vertices with no edges between them. It follows that for every forest H there is c > 0 so that if G contains neither H nor its complement as …

Close relatives (of Feedback Vertex Set), revisited

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a $2^{o(k \log k)} \cdot n^{\mathcal{O}(1)}$-time algorithm on graphs of treewidth at most $k$, assuming the Exponential Time Hypothesis. This contrasts with the $3^{k} \cdot k^{\mathcal{O}(1)} \cdot n$-time algorithm for FVS using the Cut&Count technique. During their live talk at IPEC 2020, Bergougnoux et al.~posed a number of open questions, which we answer in this work. - Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Sub…

Convex subgraphs and spanning trees of the square cycles

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.

有向平方环中的生成树

Spanning trees in directed square cycles

本文研究了有向平方环中的生成树结构，推导了其数量公式，并揭示了其与雅各布斯数的关系。

在有向图之间最大化同态计数

Maximising homomorphism counts between digraphs

本文证明了有向树的Sidorenko型不等式，表明在有向图中，纯入星和出星最大化同态计数，方法基于迭代叶重分配和Hölder不等式。

对称差（简并度）与带符号树模型

Symmetric-Difference (Degeneracy) and Signed Tree Models

本研究提出sd-简并度这一拓展对称差的图参数，证明其不具遗传性且覆盖更广图类，设计了最优的$\tilde{O}(\n)$比特邻接标记方案，并给出相关判定问题的复杂性结果。

单模随机树的谱原子

Spectral atoms of unimodular random trees

我们利用质量传输原理分析了么模随机树邻接算子的预解式$(A-z)^{-1}$的局部递归。在复杂的参数$z$接近一个给定的位置$λ$的实轴上的限制，我们表明，这种递归诱导分解的树成有限块的几何形状直接确定的光谱质量在$λ$。然后，我们利用这种对应关系，以获得精确的信息的纯点支持的频谱，在扩展性能的树。特别地，我们推导出任何具有最小度$δ\ge 3$和最大度$Δ$的幺模随机树的谱的纯点支撑被限制为100个点，即大小小于$\frac{Δ-2}{δ-2}$的树的特征值。更一般地说，我们表明，限制$δ\ge 3$可以被削弱到$δ\ge 2$，只要锚定的等周常数的树保持有界远离$0$。这特别适用于任何没有叶子的幺模高尔顿-沃森树，允许我们解决Bordenave，Sen和Virág（2013）的猜想。