Answers to Two Questions on the DP Color Function

Abstract

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and 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 $G$. In 2019, Kaul and Mudrock 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 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 $p=1$.

DP color functions versus chromatic polynomials

“…By the given condition, ℓ G (e) ≥ 3 is odd for each e ∈ E(G) \ E(T ), implying that X r = ∅ for each even r ≥ 4. If X 3 = ∅, the result can actually be proved by the idea in the proof of Theorem 7 in [11].…”

“…Let G ∨ H denote the join of G and H for any two vertex disjoint graphs G and H. Recently, Mudrock and Thomason[11] showed that P DP (K 1 ∨ G) ≈ P (K 1 ∨ G) for each graph G. Their result answered a question asked by Kaul and Muddock[10]: for every graph G, does there exists p ∈ N such that P DP (K p ∨ G) ≈ P (K p ∨ G)? Note that a graph has a dominating vertex (i.e., a vertex in the graph which is adjacent to all other vertices in the graph) if and only if it is the graph H ∨ K 1 for some graph H.…”

DP-Coloring Cartesian Products of Graphs

“…As the DP color function of an even cycle demonstrates, unlike the list color function, the DP color function need not be equal to the chromatic polynomial even for sufficiently large values of m. In fact, Dong and Yang [6] recently (extending results of [13]) showed that if G contains an edge e such that the length of a shortest cycle containing e in G is even, then there exists N ∈ N such that P DP (M, m) < P (M, m) whenever m ≥ N . In general, it was shown in [18] that for every n-vertex graph G, P (G, m) − P DP (G, m) = O(m n−3 ) as m → ∞; it follows that for any graph G whose P DP (G, m) is a polynomial in m for large enough m, the polynomial will have the same three terms of highest degree as P (G, m).…”

“…Section 3 begins with definitions of canonical labeling and twisted-canonical labeling of covers which give a characterization of the bad covers of odd and even cycles respectively. These notions of labelings are of independent interest in the study of DP coloring (see e.g., [1,10,18]). Using these tools along with volatile coloring, we end Section 3 by showing that Theorem 4 is sharp when G is an even cycle and k = 1; that is, for any m ∈ N, f (C 2m+2 , 1) = P DP (C 2m+2 , 3) = 2 2m+2 − 1.…”

The DP Color Function of Clique-Gluings of Graphs

“…Two of the current authors (Kaul and Mudrock in [12]) introduced a DP-coloring analogue of the chromatic polynomial to gain a better understanding of DP-coloring and use it as a tool for making progress on some open questions related to the list color function. Since its introduction in 2019, the DP color function has received some attention in the literature (see [1,5,7,11,12,13,16,17]). Suppose H = (L, H) is a cover of graph G. Let P DP (G, H) be the number of H-colorings of G. Then, the DP color function of G, P DP (G, m), is the minimum value of P DP (G, H) where the minimum is taken over all possible m-fold covers H of G. 3 It is easy to show that for any graph G and m ∈ N, P DP (G, m) ≤ P ℓ (G, m) ≤ P (G, m).…”