Shameful Inequalities for List and DP Coloring of Graphs

TL;DR

This paper investigates inequalities related to list and DP coloring functions of graphs, extending known results about chromatic polynomials and revealing cases where these inequalities hold or fail.

Contribution

The authors prove that certain inequalities analogous to Dong's for chromatic polynomials hold for list and DP color functions, and identify conditions where dual DP inequalities fail or hold.

Findings

Inequalities hold for list and DP color functions for all k in natural numbers.

Counterexamples show dual DP inequalities do not always hold.

Complete bipartite graphs satisfy the inequalities for dual DP color functions when k ≥ n-1.

Abstract

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 $k \in \mathbb{N}$, $P_{DP}(G, k) \leq P_\ell(G,k) \leq P(G,k) \leq P_{DP}^*(G,k)$. 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 $G$, $P(G,k+1)/(k+1)^n \geq P(G,k)/k^n$ for all $k \in \mathbb{N}$ satisfying $k \geq n-1$. 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 $G$, $P_{\ell}(G,k+1)/(k+1)^n \geq P_{\ell}(G,k)/k^n$ and $P_{DP}(G,k+1)/(k+1)^n \geq P_{DP}(G,k)/k^n$ 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 $P_{DP}^*(G,k+1)/(k+1)^n \geq P_{DP}^*(G,k)/k^n$ for all $k \in \mathbb{N}$ satisfying $k \geq n-1$ when $G$ is an $n$-vertex complete bipartite graph.