AI 中文总结
本文构造了一个40顶点图，证明爆破缺陷染色分离常数下界从30/29改进到11/10，且在最小时缺陷d=2处实现。
AI 中文摘要
对于图 $G$ 和整数 $d \geq 0$，令 $\chi^d(G)$ 表示 $d$-缺陷染色数，并令 $G \boxtimes K_{d+1}$ 为 $G$ 的 $(d+1)$ 重团爆破。Norin 和 Steiner 通过构造无穷多个 $d$ 使得存在图 $G$ 满足 $\chi(G) \geq (30/29) \chi^d(G \boxtimes K_{d+1})$，从而否证了 Guo、Kang 和 Zwaneveld 的猜想 $\chi(G) = \chi^d(G \boxtimes K_{d+1})$，并证明了通用上界 $\chi(G) \leq 2 \chi^d(G \boxtimes K_{d+1})$。记 $C_d = \sup_G \chi(G)/\chi^d(G \boxtimes K_{d+1})$ 和 $C^* = \sup_d C_d$，他们的结果给出 $C^* \in [30/29, 2]$。我们改进下界：我们构造一个显式的 40 顶点图 $W$，满足 $\chi(W) = 11$ 且 $\chi^2(W \boxtimes K_3) = 10$，从而 $C^* \geq C_2 \geq 11/10 > 30/29$，且这已经在使得这种分离可能的最小缺陷 $d = 2$ 处实现。所有参数均由证明确立；唯一的计算机辅助输入——某个 30 顶点、10 色列表实例 $(B,L)$ 的不可列表染色性——由可独立验证的 DRAT 反驳证明认证。
英文摘要
For a graph $G$ and an integer $d \geq 0$, let $χ^d(G)$ denote the $d$-defective chromatic number, and let $G \boxtimes K_{d+1}$ be the $(d+1)$-fold clique blowup of $G$. Norin and Steiner disproved the conjecture $χ(G) = χ^d(G \boxtimes K_{d+1})$ of Guo, Kang and Zwaneveld by exhibiting, for infinitely many $d$, graphs with $χ(G) \geq (30/29) χ^d(G \boxtimes K_{d+1})$, and they proved the universal upper bound $χ(G) \leq 2 χ^d(G \boxtimes K_{d+1})$. Writing $C_d = \sup_G χ(G)/χ^d(G \boxtimes K_{d+1})$ and $C^* = \sup_d C_d$, their results give $C^* \in [30/29, 2]$. We improve the lower bound: we exhibit an explicit 40-vertex graph $W$ with $χ(W) = 11$ and $χ^2(W \boxtimes K_3) = 10$, so that $C^* \geq C_2 \geq 11/10 > 30/29$, already at the smallest defect for which such a separation is possible, namely $d = 2$. All parameters are established by the proofs; the only computer-assisted input, the non-list-colourability of a certain 30-vertex, 10-colour list instance $(B,L)$, is certified by an independently checkable DRAT refutation.
Comments5 pages. Ancillary files included: DRAT non-list-colourability certificate, CNF, and verification scripts (all machine-checkable)
