Combinatorics of multivariate chromatic polynomials for rooted graphs

Abstract

Richard Stanley defined the chromatic symmetric function XG of a graph G and conjectured that trees T and U are isomorphic if and only if XT = XU . We study a variation of the chromatic symmetric function for rooted graphs, where we require the root vertex to have a specified color. In addition to presenting various combinatorial identities and recursions satisfied by these rooted chromatic polynomials, this work contains three main results. The first is that our polynomials satisfy the analogue of Stanley's conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. We give two proofs of this theorem, one based on unique factorization, the other based on algebraic transformations to pointed chromatic functions and rooted U -polynomials. The second result, which is used in the factorization-based proof, is that for each tree T , XT (x1, . . . , xN ) is irreducible in Q[x1, . . . , xN ] for large enough N . We prove this by a novel application of Eisenstein's Criterion. The third result answers a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.