Answers to two questions on the DP color function (Q2030743)

Summary: DP-coloring is a generalization of list coloring that was introduced in [J. Comb. Theory, Ser. B 129, 38--54 (2018; Zbl 1379.05034)] by \textit{Z. Dvořák} and \textit{L. Postle}. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic polynomial of graph \(G\) is denoted \(P(G,m)\), and it is equal to the number of proper \(m\)-colorings of \(G\). \textit{H. Kaul} and \textit{J. A. Mudrock} [Adv. Appl. Math. 123, Article ID 102131, 26 p. (2021; Zbl 1479.05109)] introduced an analogue of the chromatic polynomial for DP-coloring; specifically, the DP color function of graph \(G\) is denoted \(P_{DP}(G,m)\). For vertex disjoint graphs \(G\) and \(H\), suppose \(G \vee H\) denotes the join of \(G\) and \(H\). Two fundamental questions posed by Kaul and Mudrock [loc. cit.] are: (1) For any graph \(G\) with \(n\) vertices, is it the case that \(P(G,m)-P_{DP}(G,m) = O(m^{n-3})\) as \(m \rightarrow \infty \)? and (2) For every graph \(G\), does there exist \(p,N \in \mathbb{N}\) such that \(P_{DP}(K_p \vee G, m) = P(K_p \vee G, m)\) whenever \(m \geq N\)? We show that the answer to both these questions is yes. In fact, we show the answer to (2) is yes even if we require \(p=1\).