An Algebraic Approach for Counting DP-3-colorings of Sparse Graphs

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, and the list color function, $P_{\ell}(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. It follows that $P_{DP}(G,m) \le P_{\ell}(G,m) \le P(G,m)$. 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 known that the DP color function is not chromatic-adherent, but there are only two known 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 number of $\mathcal{H}$-colorings of $G$. We then use a well-known result of Alon and F\"{u}redi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on $P_{DP}(G,3)$ when $2n > |E(G)|$. An easy consequence of this is that $P_{DP}(G, 3) \geq 3^{n/6}$ for every $n$-vertex planar graph $G$ of girth at least 5, improving the previously known bounds on both $P_{DP}(G, 3)$ and $P_{\ell}(G, 3)$. Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP color function.