A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs

Elusive extremal graphs

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that `every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints.

对Lovász-Schrijver SDP层次稳定集多面体的计算搜索中的最小障碍图

A Computational Search for Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy

本文研究了通过Lovász-Schrijver SDP算子生成的稳定集多面体的lift-and-project松弛，提出LS+证书包以提升数值下界证明的透明度和可靠性，并通过计算搜索发现3-最小图和4-最小图的数量增加，揭示了极值图的新结构特征。

极小双色匹配覆盖图

Extremal minimal bipartite matching covered graphs

本文研究极小双色匹配覆盖图，证明了这类图可以通过两个无度为2的树的叶节点连接得到，并探讨了相关极小类别的特征及它们之间的关系。

正交表示、向量色数与扩展复杂度

Orthonormal representations, vector chromatic number, and extension complexity

本研究通过构建Alon和Szegedy近正交向量的二分推广，得到多个涉及Lovász theta函数、向量色数等的极值问题强界，并推导了两个具独立研究价值的向量色数通用下界。

Sharp bounds for the chromatic number of random Kneser graphs

Given positive integers $n\ge 2k$, the {\it Kneser graph} $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the {\it random Kneser graph} $KG_{n,k}(p)$, that is, the graph obtained from $KG_{n,k}$ by including each of the edges of $KG_{n,k}$ independently and with probability $p$. …