Year: 2026
Type: preprint
Abstract: Guo, Kang and Zwaneveld conjectured that colouring the \((d+1)\)-clique blowup of a graph with defect \(d\) requires as many colours as colouring the graph properly. Norin and Steiner refuted this, proved that the two chromatic numbers differ by at most a factor of two, and asked for the exact constant; the best separation known was \(11/10\). We prove that the constant is exactly two for every fixed defect \(d\ge 2\). Every rational ratio in \([1,2)\) is attained, by graphs whose clique number and fractional chromatic number equal the defective parameter: the separation is invisible to the standard lower bounds. The engine is an exact transfer theorem: a weighted list-colouring obstruction spread over an arbitrary auxiliary graph yields prescribed defective and ordinary chromatic numbers; Kneser graphs then give sharpness. A 29-vertex graph with defective parameter four and chromatic number five answers Norin's Barbados question at the smallest possible defect and palette.
Source: arXiv (Cornell University)
Author: Lecomte, Guillaume
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213505105