A counterexample to a conjecture of Björner and Lovász on the $χ$-coloring complex

Lovász猜想的无穷多反例

Infinitely many counterexamples to a conjecture of Lovász

本文针对Lovász关于通过移除r-1个顶点减小匹配数的猜想，基于广义Petersen图构造了r=3时的无穷反例族，并给出了r=4时的具体反例。

Lovász 一个猜想的反例

A Counterexample to a Conjecture of Lovász

本文构造 r=3 时 Lovász 超图猜想的反例，即具有 102 个顶点的高度对称立方图 Biggs-Smith 图的线超图，证明该猜想不成立。

Small 4-regular planar graphs that are not circle representable

A 4-regular planar graph $G$ is said to be circle representable if there exists a collection of circles drawn on the plane such that the touching and crossing points correspond to the vertices of $G$, and the circular arcs between those points correspond to the edges of $G$. Lovász (1970) conjectured that every 4-regular planar graph has a circle representation, but an infinite family of counterexamples was given by Bekos and Raftopoulou (2015). We reduce the order of the smallest known counterexamples among simple graphs from 822 to 68 based on a multigraph counterexample of order 12.

关于图不变量与Lovász $\vartheta$-函数的观察

Observations on Graph Invariants with the Lovász $\vartheta$-Function

本文利用Lovász $\vartheta$-函数研究了图的Shannon容量、同谱不同构图及图不变量界限，推导了强正则图等图类的精确色数表达式，并反例证明Schrijver变体不具备Shannon容量上界性质。

量子 Hedetniemi 猜想的一个反例

A counterexample to the quantum Hedetniemi conjecture

本文构造显式有限图，否证量子 Hedetniemi 猜想，证明其量子色数乘积小于因子最小值，并给出更小反例及 Lean 4 形式化验证。

On a generalized Erdős-Rademacher problem

The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $\lfloor n^2/4 \rfloor +t$ edges and triangle covering number $s$, we determine (for large $n$) sharp bounds on the minimum number of triangles in $G$ and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erd\H os, and Lovász-Simonovits whose results apply only to $s \le t$. Our results also address two conjectures of Xia…