An improved lower bound for the blowup defective chromatic separation constant

Abstract

For a graph $G$ and an integer $d\ge 0$, let ${\chi }^d(G)$ denote the $d$-defective chromatic number, and let $G⊠K_{d+1}$ be the $(d+1)$-fold clique blowup of $G$. Norin and Steiner disproved the conjecture $\chi (G)={\chi }^d(G⊠K_{d+1})$ of Guo, Kang and Zwaneveld by exhibiting, for infinitely many $d$, graphs with $\chi (G)\ge (30/29){\chi }^d(G⊠K_{d+1})$, and they proved the universal upper bound $\chi (G)\le 2{\chi }^d(G⊠K_{d+1})$. Writing $C_d={\sup }_G\chi (G)/{\chi }^d(G⊠K_{d+1})$ and $C^{\ast }={\sup }_d{C}_d$, their results give $C^{\ast }\in [30/29,2]$. We improve the lower bound: we exhibit an explicit 40-vertex graph $W$ with $\chi (W)=11$ and ${\chi }^2(W⊠K_3)=10$, so that $C^{\ast }\ge C_2\ge 11/10>30/29$, 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.

arXiv comment5 pages. Ancillary files included: DRAT non-list-colourability certificate, CNF, and verification scripts (all machine-checkable)