The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was introduced by Birkhoff in 1912; denoted P(G, k), it equals the number of proper k-colorings of graph G. Enumerative analogues of the chromatic polynomial of a graph have been introduced for two well-studied generalizations of ordinary coloring, namely, list colorings: \(P_{\ell }\) , the list color function (1990); and DP colorings: \(P_{DP}\) , the DP color function (2019), and \(P^*_{DP}\) , the dual DP color function (2021). For any graph G and \(k \in \mathbb {N}\) , \(P_{DP}(G, k) \le P_\ell (G,k) \le P(G,k) \le P_{DP}^*(G,k)\) . In 2000, Dong settled a conjecture of Bartels and Welsh from 1995 known as the Shameful Conjecture by proving that for any n-vertex graph G, \(P(G,k+1)/(k+1)^n \ge P(G,k)/k^n\) for all \(k \in \mathbb {N}\) satisfying \(k \ge n-1\) . In contrast, for infinitely many positive integers n, Seymour (1997) gave an example of an n-vertex graph for which the above inequality does not hold for some \(k = \Theta (n/ \log n)\) . In this paper, we consider analogues of Dong’s result for list and DP color functions. Specifically, in contrast to the chromatic polynomial, we prove that for any n-vertex graph G, \(P_{\ell }(G,k+1)/(k+1)^n \ge P_{\ell }(G,k)/k^n\) and \(P_{DP}(G,k+1)/(k+1)^n \ge P_{DP}(G,k)/k^n\) for all \(k \in \mathbb {N}\) . For the dual DP analogue of these inequalities, we show that there is a graph G and \(k \in \mathbb {N}\) such that \(P_{DP}^*(G,k+1)/(k+1)^n < P_{DP}^*(G,k)/k^n\) , and we prove \(P_{DP}^*(G,k+1)/(k+1)^n \ge P_{DP}^*(G,k)/k^n\) for all \(k \in \mathbb {N}\) satisfying \(k \ge n-1\) when G is an n-vertex complete bipartite graph.
