On Polynomial Representations of Dual DP Color Functions

TL;DR

This paper investigates polynomial representations of dual DP color functions, revealing that unlike DP color functions, dual DP color functions do not necessarily become polynomial for large values, and connects this to signed graph chromatic polynomials.

Contribution

It demonstrates that dual DP color functions do not always stabilize as polynomials for large m, contrasting with DP color functions, and links this behavior to signed graph chromatic polynomials.

Findings

Dual DP color functions are not always polynomial for large m.

Established a connection between dual DP color functions and signed graph chromatic polynomials.

Contrasted the behavior of dual DP color functions with DP color functions regarding polynomial stabilization.

Abstract

DP-coloring (also called correspondence coloring) is a generalization of list coloring that was introduced by Dvo\v{r}\'{a}k 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: $P_{\ell}$, list color functions (1990); DP colorings: $P_{DP}$, DP color functions (2019), and $P^*_{DP}$, dual DP color functions (2021). For any graph $G$ and $m \in \mathbb{N}$, $P_{DP}(G, m) \leq P_\ell(G,m) \leq P(G,m) \leq P_{DP}^*(G,m)$. In 2022 (improving on older results) Dong and Zhang showed that for any graph $G$, $P_{\ell}(G,m)=P(G,m)$ whenever $m \geq |E(G)|-1$. 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 $N \in \mathbb{N}$ and a polynomial $p(m)$ such that $P_{DP}(G,m) = p(m)$ whenever $m \geq N$? 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.