Hadwiger conjectured that every graph of chromatic number k admits a clique minor order k. Here we prove for ≤ 10, with unique k-coloring (up to the color names) The proof does not rely on Four Color Theorem.

A mixed hypergraph is a triple H = (X, C,D) where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A k-coloring of H is a mapping c : X → [k] such that each C-edge has two vertices with the same color and each D-edge has two vertices with distinct colors. H = (X, C,D) is called a mixed hypertree if there exists a tree T = (X, E) such that ...