 Vector coloring

A vector $k$-coloring, for $k>1$, assigns a unit vector $v_i$ to each vertex of a graph, such that $⟨v_i,v_j⟩\le -1/(k-1)$ on every edge. A graph colouring with $k$ colors gives a vector $k$-coloring by placing the colors at the vertices of a regular simplex. The Gram matrix of these vectors allows semidefinite programming to search for such a representation.