A defective coloring of a simple graph assigns colors to its vertices so that the maximum
vertex degree of each nonempty monochromatic vertex-induced
subgraph is at most a prescribed nonnegative
integer defect. A defect of zero gives a proper
coloring. A clustered coloring with monochromatic
connected components of size at most a positive integer 
has defect at most 
, but a bound on the defect need not bound the sizes of monochromatic
connected components (Grzesik et al. 2026).
Defective Coloring
See also
Clustered Coloring, Maximum Average Degree, Proper Coloring, Vertex ColoringExplore with Wolfram|Alpha
References
Grzesik, A.; Kopfová, L.; Kucheriya, G.; Li, B.; and Prorok, M. "Proof of the mad Conjecture and Its Coloring Applications." 3 Oct 2026. https://arxiv.org/abs/2610.04645.Cite this as:
Weisstein, Eric W. "Defective Coloring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DefectiveColoring.html