On Polynomial Representations of Dual DP Color Functions

DP-coloring (also called correspondence coloring) is a generalization of list coloring that was introduced by Dvor\'ak and Postle in 2015. The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was introduced by Birkhoff in 1912; denoted P (G, m), it equals the number of proper m-colorings of graph G. Counting function analogues of chromatic polynomials have been introduced for list colorings: P_, list color functions (1990) ; DP colorings: P₃₏, DP color functions (2019), and P^*₃₏, dual DP color functions (2021). For any graph G and m N, P₃₏ (G, m) P_ (G, m) P (G, m) P₃₏^* (G, m). In 2022 (improving on older results) Dong and Zhang showed that for any graph G, P_ (G, m) =P (G, m) whenever m |E (G) |-1. Consequently, the list color function of a graph is a polynomial for sufficiently large m. One of the most important and longstanding open questions on DP color functions asks: for every graph G is there an N N and a polynomial p (m) such that P₃₏ (G, m) = p (m) whenever m N? We show that the answer to the analogue of this question for dual DP color functions is no. Our proof reveals a connection between a dual DP color function and the balanced chromatic polynomial of a signed graph introduced by Zaslavsky in 1982.