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An Algebraic Approach to the Non-chromatic Adherence of the DP Color Function

Samantha L. DahlbergHemanshu KaulJeffrey A. Mudrock
Dec 2022
摘要
DP-coloring (or correspondence coloring) is a generalization of list coloringthat has been widely studied since its introduction by Dvo\v{r}\'{a}k andPostle in 2015. As the analogue of the chromatic polynomial of a graph $G$,$P(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts theminimum number of DP-colorings over all possible $m$-fold covers. A function$f$ is chromatic-adherent if for every graph $G$, $f(G,a) = P(G,a)$ for some $a\geq \chi(G)$ implies that $f(G,m) = P(G,m)$ for all $m \geq a$. It is knownthat the DP color function is not chromatic-adherent, but there are only twoknown graphs that demonstrate this. Suppose $G$ is an $n$-vertex graph and$\mathcal{H}$ is a 3-fold cover of $G$, in this paper we associate with$\mathcal{H}$ a polynomial $f_{G, \mathcal{H}} \in \mathbb{F}_3[x_1, \ldots,x_n]$ so that the number of non-zeros of $f_{G, \mathcal{H}}$ equals the numberof $\mathcal{H}$-colorings of $G$. We then use a well-known result of Alon andF\"{u}redi on the number of non-zeros of a polynomial to establish anon-trivial lower bound on $P_{DP}(G,3)$ when $2n > |E(G)|$. Finally, we usethis bound to show that there are infinitely many graphs that demonstrate thenon-chromatic-adherence of the DP color function.
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