Proof of `Neighborhood covers of weak coloring degree`
What this proof establishes
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
Neighborhood covers of weak coloring degree (Theorem 6.2 of Grohe–Kreutzer–Siebertz, via their Lemma 6.9): every graph has, for every radius , an -neighborhood cover of radius whose degree is at most its weak -coloring number.
Proof strategy
Choose an ordering attaining : the defining infimum is over a nonempty set of natural-number bounds, so it is attained. Apply to that ordering and its bound. The arbitrary-order construction is discharged by above.