A rooted variant of Stanley's chromatic symmetric function

Richard Stanley defined the chromatic symmetric function X-G of a graph G and asked whether there are non-isomorphic trees T and U with X-T=X-U. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted U-polynomials, and prove three main theorems. First, for all non-empty connected graphs G, Stanley's polynomial X-G(x(1),& mldr;,x(N)) is irreducible in Q[x(1),& mldr;,x(N)] for all large enough N. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting x(0)=x(1)=q, x(2)=x(3)=1, and x(n)=0 for n>3) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.