contraction clique number | homomorphism degree

The Hadwiger number of a graph G, variously denoted η(G) (Zelinka 1976, Ivančo 1988) or h(G), is the number of vertices in the largest complete minor of G . The Hadwiger number is also called the contraction clique number or homomorphism degree. The Hadwiger conjecture states that for any loopless graph G, h(G)>=χ(G), where χ(G) the chromatic number of G. Forests have h<=2, graphs with treewidth at most 2 have h<=3, planar graphs have h<=4, and apex graphs have h<=5. Computing the Hadwiger number of a graph is an NP-hard problem.

graph minor | Hadwiger-Nelson problem