On Chromatic Functors and Stable Partitions of Graphs

Abstract

Abstract. e chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two nite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. e key ingredient in the proof is the use of stable partitions of graphs. e latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our rst result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. e second result is that the composition of the chromatic functor associated to a nite graph restricted to the category FI of nite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb.