The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k-colorability of a graph G is equivalent to the condition 1 ∈ I G,k for a certain ideal I G,k ⊆ k[x 1 , . . . , x n ]. In this paper, we extend this result by proving a general decomposition theorem for I G,k . This theorem allows us to give an algebraic characterization of uniquely k-colorable graphs. Our results also give algorithms for testing unique colorability. As an application, we verify a counterexample to a conjecture of Xu concerning uniquely 3-colorable graphs without triangles.

Abstract. Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K, and let S X be the symmetric group of X. The group S X acts naturally on R, and this in turn gives R the structure of a module over the group ring R[S X ]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have any number of minimal generators, answering this question positively.Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K. Write S X (resp. S N ) for the symmetric group of X (resp. {1, . . . , N}) and R[S X ] for its (left) group ring, which acts naturally on R.Aschenbrenner and Hillar recently proved [1] that all symmetric ideals are finitely generated over R[S X ]. They were motivated by finiteness questions in chemistry [3] and algebraic statistics [2]. In proving the Noetherianity of R, it was shown that a symmetric ideal I has a special, finite set of generators called a minimal Gröbner basis. However, the more basic question of whether I is always cyclic (already asked by Josef Schicho [4]) was left unanswered in [1]. Our result addresses a generalization of this important issue. Theorem 1. For every positive integer n, there are symmetric ideals of R generated by n polynomials which cannot have fewer than n R[S X ]-generators.In what follows, we work with the set X = {x 1 , x 2 , x 3 , . . .}, although as remarked in [1], this is not really a restriction. In this case, S X is naturally identified with S ∞ , the permutations of the positive integers, and σx i = x σi for σ ∈ S ∞ .Let M be a finite multiset of positive integers and let i 1 , . . . , i k be the list of its distinct elements, arranged so thatFor instance, the multiset M = {1, 1, 1, 2, 3, 3} has type λ(M ) = (3, 2, 1). Multisets are in bijection with monomials of R. Given M , we can construct the monomial:Conversely, given a monomial, the associated multiset is the set of indices appearing in it, along with multiplicities. The action of S ∞ on monomials coincides with the