On distinguishing special trees by their chromatic symmetric functions

Abstract: In 1995, Stanley introduced the well-known chromatic symmetric function $X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric functions such that for each vertex coloring of $G$ there is exactly one of these summands. The question, whether $X_{G}(x_1,x_{2},\ldots)$ distinguishes nonisomorphic trees with the same number of vertices, is still open in general. For special trees it has already been shown. In 2008, Martin, Morin and Wagner proved it for spiders and some caterpillars. We decompose a tree by separating the leafs and their neighbors and do the same to the remaining forest until there remains a forest with vertices of degree not greater than $1$. For nonisomorphic trees $G$ and $H$ with the same number of vertices and special properties concerning their number of leafs and leaf neighbors for each subgraph in their leaf decomposition we prove that the chromatic symmetric function distinguishes the graphs. Our idea is to find independent partitions of $G$ and $H$ with respectively a block of maximal cardinality for $G$ as well as for $H$. These cardinalities are different for graphs $G$ and $H$ with special properties of their leaf decompositions. Additionally, we give explicit formulas for the cardinality of such a maximal block of an independent partition of star connections and spiders.