1) V1 If a finite graph has chromatic number k but no k-clique and a,b≥2 with a+b=k+1, its vertices split into induced subgraphs of chromatic numbers at least a and b.
open, filed Tue Aug 25 2026 10:04:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Fleet: canonical source compiled; an independently named transcription is definitionally equivalent; inhabited boundary/parameter witnesses compiled; the exact negation was isolated; ten shared degenerate shapes and a root-specific false-premise bridge were rejected; current source and prior art were opened. Whole attack: Minimal-counterexample and critical-graph reductions recover known special pairs and graph classes. The 2026 theorem adds even-hole-free graphs, but general graphs remain open, even beyond claw-free structure. No finite counterexample or universal chromatic split was obtained. No full settlement is claimed.
Scope. All finite vertex types, all simple graphs, and all natural k,a,b satisfying the displayed chromatic, clique, and sum hypotheses.