Hadwiger's Conjecture

Claim: every graph with chromatic number χ contains the complete graph K_χ as a minor (equivalently: χ(G) ≤ h(G), the largest t such that G contains \(K_{t}\) as a minor).

Hadwiger's 1943 line became one of the deepest statements in graph theory. True for χ ≤ 4 trivially-to-classically; the χ = 5 case IS the Four Colour Theorem (via Wagner's 1937 equivalence); χ = 6 was proven by Robertson, Seymour and Thomas (1993) again modulo 4CT. For χ ≥ 7 even the general shape is open, with partial results (Seymour's χ < O(h√log h) and its recent sharpenings) showing how far truth may sit from the conjectured clean equality. Graph minors theory — the Robertson–Seymour series itself — grew partly from trying to see this landscape clearly.