具有对数周长的色-临界图

小型边集族的短彩虹圈

Short rainbow cycles for families of small edge sets

本文扩展相关结果，证明小比例非星型颜色类即可保证图的对数级彩虹围长，还确定了彩虹围长从线性过渡到对数级的颜色类类型阈值比例。

Circular chromatic number of signed graphs

A signed graph is a pair $(G, σ)$, where $G$ is a graph and $σ: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, σ)$ a circular $r$-coloring of $(G, σ)$ is an assignment $ψ$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $σ(e)=+$, then $ψ(u)$ and $ψ(v)$ have distance at least $1$, and if $σ(e)=-$, then $ψ(v)$ and the antipodal of $ψ(u)$ have distan…

超图的Hajnal-Szemeredi定理

Toward a Hajnal-Szemeredi theorem for hypergraphs

设$H$是一个具有最大度$d>1$的三元系，设$r>10^7\sqrt{d}\log^{2}d$ .然后$H$有一个适当的顶点着色与$r$颜色，使任何两个颜色类的大小相差最多一个。$r$上的结合在除了对数因子之外的数量级上是尖锐的。此外，这样的$r$ -着色可以通过随机化算法找到，该算法的预期运行时间是$\cH$的顶点数的多项式。这是将Hajnal-Szemerédi定理推广到超图的第一个结果。

Lower bounds for piercing and coloring boxes

Given a family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, let $τ$ denote its piercing number, and $ν$ its independence number. It is an old question whether $τ/ν$ can be arbitrarily large for given $d\geq 2$. Here, for every $ν$, we construct a family of axis-parallel boxes achieving $$τ\geq Ω_d(ν)\cdot\left(\frac{\log ν}{\log\log ν}\right)^{d-2}.$$ This not only answers the previous question for every $d\geq 3$ positively, but also matches the best known upper bound up to double-logarithmic factors. Our main construction has further implications about the Ramsey and coloring prop…

诱导子图与树分解 XV：具有有界团数的无偶洞图具有对数树宽

Induced subgraphs and tree decompositions XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth

本文证明了无偶洞图在团数有界时树宽至多为对数级，解决了 Sintiari 和 Trotignon 的猜想，并由此为若干 NP-难问题在该图类上给出多项式时间算法。

Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graph…

具有有界诱导圈填充数的稀疏图具有对数树宽

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

该研究证明了不含k个两两不交且不相邻圈的稀疏图具有对数级树宽，且该界是紧的，由此推出这类图上多个核心NP完全问题可在多项式或拟多项式时间内求解。

Color-avoiding percolation of random graphs: between the subcritical and the intermediate regime

Fix a graph $G$ in which every edge is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ are CA-connected if $u$ and $v$ may be connected using any subset of $k - 1$ colors. CA-connectivity is an equivalence relation dividing the vertex set into classes called CA-components. In two recent papers, Ráth, Varga, Fekete, and Molontay, and Lichev and Schapira studied the size of the largest CA-component in a randomly colored random graph. The second of these works distinguished and studied three regimes (supercritical, intermediate, and subcritical) in which the largest CA-component has …

导出子图与树分解 X：迈向偶洞无图的对数树宽

Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs

本文证明了无大团且无广义 $t$-棱锥导出子图的偶洞无图树宽至多为 $c(t)\log{n}$，解决了 Sintiari-Trotignon 猜想特例，并使多个 NP-难问题在此类图上有多项式时间算法。

Conflict-Free Coloring of String Graphs

Conflict-free coloring (in short, CF-coloring) of a graph $G = (V,E)$ is a coloring of $V$ such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane: (i) We provide a general upper bound of $O(χ(G)^2 \log n)$ on the CF-chromatic number of any string graph $G$ with $n$ vertice…

约束Ramsey数

Constrained Ramsey Numbers

对于两个图S和T，约束Ramsey数f（S，T）是最小的n，使得n个顶点上的完全图的任意颜色数的边染色都有一个与S同构的单色子图或一个与T同构的彩虹子图（所有边的颜色不同）. Erdos-Rado正则Ramsey定理暗示f（S，T）存在当且仅当S是星或T是无圈的，并且已经做了很多工作来确定f（S，T）对各种类型参数的增长率。当S和T都是分别有s和t条边的树时，Jamison，Jiang，and Ling证明了f（S，T）<= O（st^2），并证明了它总是至多O（st）。本文研究了T是一条路的情形，证明了f（S，P_t）= O（st log t），这与猜想仅相差一个对数因子。这实质上改进了大多数s和t值的先前边界。

Erdős-Rényi 随机图上的避色渗流

Color-avoiding percolation on the Erdős-Rényi random graph

本文研究常数平均度的随机染色 Erdős-Rényi 随机图中的避色渗流，刻画最大 CA 分量呈线性、对数及有界大小的三种状态。

小型偶覆盖、局部可译码与边着色Kikuchi图的受限子图

Small Even Covers, Locally Decodable Codes and Restricted Subgraphs of Edge-Colored Kikuchi Graphs

针对k一致超图中偶覆盖与超边数量的极值权衡问题，提出纯组合论证恢复偶数k的已知最佳界，引入Kikuchi图新变体改进奇数k的对数因子，并给出3查询二进制线性局部可译码改进下界的纯组合证明。

$p$-中心染色的两个下界

Two lower bounds for $p$-centered colorings

本文针对$p$-中心染色问题，构造了浅次图平均度多项式有界的单调图类，证明其所需颜色数超多项式于$p$；并证明最大度$\Delta$的图存在下界$\Omega(\Delta^{2-1/p} p \ln^{-1/p}\Delta)$，匹配已知上界至对数因子。

舒尔型数的精确上界

Refined upper bounds on Schur-like numbers

该研究针对舒尔型数，得出当N满足3^r (r!)^{1/m}时的精确上界，且当m与r成对数关系时该上界具定性最优性，解决了相关组合数论问题。