Euler characteristics and chromatic polynomials (Q2372417)

This work studies the relation between the chromatic polynomial of a graph \(G\) and the Euler characteristic of certain spaces. These spaces are obtained by a construction which is a generalization of the configuration space. The authors show, in the case that \(G\) has only one point, the following theorem: Let \(G\) be a graph and \(M_G\) the generalized configuration space with \(M=CP^{\lambda-1}\). Then \(c(G, \lambda)=\chi(M_G)\). Then some graph-theoretic results are given. For example the authors show: Let \(K_p\) denote the complete graph on \(p\) vertices. Suppose that \(G=G_1\cup G_2\) and \(G_1\cap G_2=K_p\). Then the chromatic polynomials satisfy \[ c(G,\lambda)= c(G_1,\lambda)\cdot c(G_2,\lambda)/ c(K_p,\lambda). \] At the end they explore the relationship of their work and other approaches. The Leray sequence associated with a pair \((X,Y)\) where \(Y\) is a submanifold of a manifold \(X\) plays an important rôle.