graph with Hadwiger number should have a graph coloring with at most colors. The case is equivalent (by Wagner's characterization of the graphs with this Hadwiger number) to the four color theorem on colorings of planar graphs, and the conjecture has also been proven for , but remains unproven for larger values of . Because of their low degeneracy, the graphs with Hadwiger number at most can be colored by a greedy coloring algorithm using colors. Computational complexity Testing whether the Hadwiger number of a given graph is at least a given value is NP-complete, from which it follows that determining the Hadwiger number is NP-hard. However, the problem is fixed-parameter tractable: there is an algorithm for finding the largest clique minor in an amount of time that depends only polynomially on the size of the graph, but exponentially in . Additionally, polynomial time algorithms can approximate the Hadwiger number significantly more accurately than the best polynomial-time approximation (assuming P ≠ NP) to the size of the largest complete subgraph. Related concepts The achromatic number of a graph is the size of the largest clique that can be formed by contracting a family of independent sets in . Uncountable clique minors in infinite graphs may be characterized in terms of havens, which formalize the evasion strategies for certain pursuit–evasion games: if the Hadwiger number is uncountable, then it equals the largest order of a haven in the graph. Every graph with Hadwiger number has at most cliques (complete subgraphs). defines a class of graph parameters that he calls -functions, which include the Hadwiger number. These functions from graphs to integers are required to be zero on graphs with no edges, to be minor-monotone, to increase by one when a new vertex is added that is adjacent to all previous vertices, and to take the larger value from the two subgraphs on either side of a clique separator. The set of all such functions forms a complete lattice under the operations of elementwise minimization and maximization. The bottom element in this lattice is the Hadwiger number, and the top element is the treewidth. Footnotes Notes References . . . . . . . . . . . . Graph invariants Graph minor theory