---
tag: EBNHD7FQ23
kind: concept
name: Repair radii and villainy
msc: [05-XX]
sources:
  - arxiv: 2610.13556v1
    locator: Definition 2.2
    mention: ax://paper/2610.13556#L0DK
    licence: cc-by
    note: the paper that states the item (corpus arxiv AX9.2)
history:
  notes: The source is arXiv:2610.13556v1 (2026-10), by Jiang Jiang, Yuhan Jiang, Yiyang Li, Lujia Wang, Aitao Zhang. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it.
inventory: [GY5XXNYENM]
uses: [0874138YP6, 07TXYYVYZR]
---

## Summary

A repair radius measures how far a colouring of a graph $G$ with a given profile of class sizes can be from a proper one.

The radius $R_G(m)$ is the largest Hamming [[distance from a point to a set|07TXYYVYZR]], over colourings with profile $m$, to the set of [[proper colourings|0874138YP6]] with profile $m$.

The weak radius $R^w_G(m)$ lets the nearest proper colouring have any profile.

The villainy $B(G)$ and the weak villainy $B_w(G)$ are the maxima of these radii over feasible profiles, and both are zero for the empty graph.