AI 中文总结
本文针对所有二分图，证实了DP着色函数的一个基本开放问题，得出其DP着色函数的显式表达式，并给出围长为偶数的二分图的P(G,q)-P_{DP}(G,q)的渐近公式。
AI 中文摘要
DP着色（又称对应着色）是列表着色的推广，自2015年由Dvořák和Postle提出以来已被广泛研究。作为图G的色多项式P(G,q)的类似物，图G的DP着色函数记为P_{DP}(G,q)，它计数G的所有q重覆盖上的最小DP着色数。由此可得P_{DP}(G,q) ≤ P(G,q)。已知存在这样的图：对所有足够大的q，都有P_{DP}(G,q) < P(G,q)；实际上，所有含环的二分图都具有该性质。关于DP着色函数的一个基本开放问题是：对每个图G，是否存在自然数N和多项式p，使得当q ≥ N时，P_{DP}(G,q) = p(q)？本文对所有二分图给出了该问题的肯定回答。具体而言，若G是含n个顶点、c个连通分支的二分图，则对所有足够大的q，有P_{DP}(G,q) = (-1)^{n-c}q^c T_G(1-q,1)，其中T_G(x,y)是G的Tutte多项式。本文提出的思路还给出了当G的围长为偶数时，P(G,q)-P_{DP}(G,q)的渐近公式。
英文摘要
DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of $P(G,q)$, the chromatic polynomial of a graph $G$, the DP color function of $G$, denoted by $P_{DP}(G,q)$, counts the minimum number of DP-colorings over all $q$-fold covers of $G$. It follows that $P_{DP}(G,q) \leq P(G,q)$. It is known that there are graphs for which $P_{DP}(G,q) < P(G,q)$ for all sufficiently large $q$; in fact, all bipartite graphs containing a cycle have this property. A fundamental open question about DP color functions asks whether, for every graph $G$, there exist $N \in \mathbb{N}$ and a polynomial $p$ such that $P_{DP}(G,q) = p(q)$ whenever $q \geq N$. In this paper we answer this question affirmatively for all bipartite graphs. Specifically, if $G$ is an $n$-vertex bipartite graph with $c$ components, then $P_{DP}(G,q) = (-1)^{n-c}q^c \;T_G(1-q,1)$ for all sufficiently large $q$, where $T_G(x,y)$ is the Tutte polynomial of $G$. The ideas we develop also yield an asymptotic formula for $P(G,q)-P_{DP}(G,q)$ whenever the girth of $G$ is even.
Comments17 pages, 1 figure
