Colin de Verdière's invariant is a graph parameter
Contents
Definition
Let
Characterization of known graph families
Several well-known families of graphs can be characterized in terms of their Colin de Verdière invariants:
These same families of graphs also show up in connections between the Colin de Verdière invariant of a graph and the structure of its complement graph:
Graph minors
A minor of a graph is another graph formed from it by contracting edges and by deleting edges and vertices. The Colin de Verdière invariant is minor-monotone, meaning that taking a minor of a graph can only decrease or leave unchanged its invariant:
If H is a minor of G thenBy the Robertson–Seymour theorem, for every k there exists a finite set H of graphs such that the graphs with invariant at most k are the same as the graphs that do not have any member of H as a minor. Colin de Verdière (1990) lists these sets of forbidden minors for k ≤ 3; for k = 4 the set of forbidden minors consists of the seven graphs in the Petersen family, due to the two characterizations of the linklessly embeddable graphs as the graphs with μ ≤ 4 and as the graphs with no Petersen family minor.
Chromatic number
Colin de Verdière (1990) conjectured that any graph 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 Robertson, Seymour & Thomas (1993) of the Hadwiger conjecture for K6-minor-free graphs.
Other properties
If a graph has crossing number k, it has Colin de Verdière invariant at most k + 3. For instance, the two Kuratowski graphs K5 and K3,3 can both be drawn with a single crossing, and have Colin de Verdière invariant at most four.
Influence
Colin de Verdière invariant is defined from a special class of matrices corresponding to a graph instead of just a single matrix related to the graph. Along the same line other graph parameters are defined and studied such as minimum rank of a graph, minimum semidefinite rank of a graph and minimum skew rank of a graph.