Graphs with no K_{s+1}-subdivision are clustered-colorable with O(s) colors, and several stronger variants hold under almost (≤1)-subdivision and minor-free assumptions.
On the conjecture of Hajós.Combina- torica, 1(2):141–143, 1981
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Clustered Variants of Haj\'os' Conjecture
Graphs with no K_{s+1}-subdivision are clustered-colorable with O(s) colors, and several stronger variants hold under almost (≤1)-subdivision and minor-free assumptions.