AI 中文总结
该研究针对不含K₅、Q₆子式的图，证明其恰当无冲突着色的7色上界，改进平面图此前的8色上界，还推导了不含Kₖ₊₁子式图的奇偶无冲突着色的2k-1色相关结论。
AI 中文摘要
恰当无冲突着色是一种恰当顶点着色，其中每个非孤立顶点的开邻域内存在唯一出现的颜色。我们证明，既不含K₅子式也不含Q₆子式（Q₆=K₃∨$\boldsymbol{\bar{K}_3}$）的所有图都可使用至多7种颜色实现此类着色，这尤其改进了平面图此前8种颜色的通用上界。证明结合了先前开发的迭代距离3选择器构造与通用锚点收缩提升原理：前者在闭邻域中提供独立着色的见证，后者将这些见证与合适子式的恰当着色结合。我们还开发了第一种机制的奇偶类似物，证明当Hadwiger猜想的Kₖ₊₁情形成立时，每个不含Kₖ₊₁子式的图都可使用2k-1种颜色进行恰当顶点着色，使得每个非孤立顶点的开邻域内存在出现奇数次的颜色。
英文摘要
A proper conflict-free coloring is a proper vertex coloring in which every nonisolated vertex has a color occurring uniquely in its open neighborhood. We prove that every graph with neither a $K_5$-minor nor a $Q_6$-minor admits such a coloring with at most seven colors, where $Q_6=K_3\vee\overline{K_3}$. In particular, this improves the previous general upper bound of eight for planar graphs. The proof combines a previously developed iterated distance-three selector construction with a general anchor-contraction lifting principle. The first supplies independently colored witnesses in closed neighborhoods, while the second combines those witnesses with a proper coloring of a suitable minor. We also develop the parity analogue of the first mechanism and show that, whenever the $K_{k+1}$ case of Hadwiger's conjecture holds, every $K_{k+1}$-minor-free graph can be proper vertex colored with $2k-1$ colors such that every nonisolated vertex has a color occurring an odd number of times in its open neighborhood.
