Hadwiger number h(G): largest t with K_t minor in G. Hadwiger's Conjecture (1943): χ(G) ≤ h(G). Implies Four Color Theorem! Proven for t≤6. One of the deepest conjectures in graph theory. Wagner: t≤4 equivalent to 4CT.
How to Use
Select graph:
- h: Hadwiger number
- Minor: K_t found
- χ ≤ h: Conjecture check
Hadwiger's Conjecture
χ(G) ≤ h(G): chromatic number bounded by Hadwiger number. t=1,2: trivial. t=3: Dirac (1952). t=4: equivalent to 4CT (Wagner). t=5: equivalent to 4CT (Wagner). t=6: Robertson-Seymour-Thomas (1993). t≥7: OPEN!
Computation
NP-hard in general (testing if K_t is a minor). For specific graphs: h(K_n)=n, h(planar)≤5 (by 4CT+Hadwiger equiv), h(K_{n,n})=n+1. Upper bounds from separator theorems. Lower bounds from graph minors.
Step-by-Step Instructions
- 1Select graph.
- 2Compute h(G).
- 3Find K_t minor.
- 4Check χ ≤ h.
- 5Apply bounds.