A blow-up construction and graph coloring

Details A blow-up construction and graph coloring A blow-up construction and graph coloring

1994-01-25

arxiv.org/abs/alg-geom/9401007v1

Discrete Math. 145 (1995), no. 1-3, 11-35

Mathematics

Algebraic Geometry

22 pages, amstex 2.1

Scientific paper

10.1016/0012-365X(94)00051-J

Given a graph G (or more generally a matroid embedded in a projective space), we construct a sequence of varieties whose geometry encodes combinatorial information about G. For example, the chromatic polynomial of G (giving at each m>0 the number of colorings of G with m colors, such that no adjacent vertices are assigned the same color) can be computed as an intersection product between certain classes on these varieties, and other information such as Crapo's invariant find a very natural geometric counterpart. The note presents this construction, and gives `geometric' proofs of a number of standard combinatorial results on the chromatic polynomial.

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