with Colin de Verdière invariant μ may be colored with at most μ + 1 colors. For instance, the linear forests have invariant 1, and can be 2-colored; the outerplanar graphs have invariant two, and can be 3-colored; the planar graphs have invariant 3, and (by the four color theorem) can be 4-colored. For graphs with Colin de Verdière invariant at most four, the conjecture remains true; these are the linklessly embeddable graphs, and the fact that they have chromatic number at most five is a consequence of a proof by of the Hadwiger conjecture for K6-minor-free graphs. Other properties If a graph has crossing number , it has Colin de Verdière invariant at most . For instance, the two Kuratowski graphs and can both be drawn with a single crossing, and have Colin de Verdière invariant at most four. Influence The Colin de Verdière invariant is defined through a class of matrices corresponding to the graph instead of just a single matrix. Along the same lines other graph parameters can be defined and studied along the same lines, such as the minimum rank, minimum semidefinite rank and minimum skew rank. Notes References . Translated by Neil J. Calkin as . . . . Graph invariants Graph minor theory