Hadwiger Conjecture (graph Theory)

Hadwiger Conjecture (graph Theory)

In graph theory, the Hadwiger conjecture (or Hadwiger's conjecture) states that, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single supervertex produces a complete graph K_k on k vertices as a minor of G.

This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. Bollobás, Catlin & Erdős (1980) call it “one of the deepest unsolved problems in graph theory.”