An improved lower bound for the blowup defective chromatic separation constant

RESUMO

For a graph $G$ and an integer $d \geq 0$, let $\chi^d(G)$ 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 $\chi(G) = \chi^d(G \boxtimes K_{d+1})$ of Guo, Kang and Zwaneveld by exhibiting, for infinitely many $d$, graphs with $\chi(G) \geq (30/29) \chi^d(G \boxtimes K_{d+1})$, and they proved the universal upper bound $\chi(G) \leq 2 \chi^d(G \boxtimes K_{d+1})$. Writing $C_d = \sup_G \chi(G)/\chi^d(G \boxtimes K_{d+1})$ 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 $\chi(W) = 11$ and $\chi^2(W \boxtimes K_3) = 10$, 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.