Defective coloring of blowups
Sergey Norin, Raphael Steiner
TL;DR
The paper studies the relationship between the $d$-defective chromatic number of the blowup $G\boxtimes K_{d+1}$ and the ordinary chromatic number $\chi(G)$. It disproves the conjecture $\chi(G)=\chi^d(G\boxtimes K_{d+1})$ by constructing graphs with $\chi^d(G\boxtimes K_{d+1})$ bounded by $k=2d^3+2d^2+d+3$ while $\chi(G)$ exceeds $k$ for infinitely many $d$, showing gaps can be arbitrarily large for $d\ge2$. On the positive side, it proves a universal two-sided bound $\chi(G)\le 2\chi^d(G\boxtimes K_d)\le 2\chi^d(G\boxtimes K_{d+1})$, limiting the discrepancy. The proofs combine the Bohman–Holzman list-coloring framework with Haxell's independent transversal theorem, yielding corollaries such as infinite families with a ratio at least $30/29$ and clarifying the tightness of the two-sided bound.
Abstract
Given a graph $G$ and an integer $d\ge 0$, its $d$-defective chromatic number $χ^d(G)$ is the smallest size of a partition of the vertices into parts inducing subgraphs with maximum degree at most $d$. Guo, Kang and Zwaneveld recently studied the relationship between the $d$-defective chromatic number of the $(d+1)$-fold (clique) blowup $G\boxtimes K_{d+1}$ of a graph $G$ and its ordinary chromatic number, and conjectured that $χ(G)=χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. In this note we disprove this conjecture by constructing graphs $G$ of arbitrarily large chromatic number such that $χ(G)\ge \frac{30}{29}χ^d(G\boxtimes K_{d+1})$ for infinitely many $d$. On the positive side, we show that the conjecture holds with a constant factor correction, namely $χ^d(G\boxtimes K_{d+1})\le χ(G)\le 2χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$.