On the List Color Function Threshold

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\mathrm{\ell }}(G,m)$, is a list analogue of the chromatic polynomial that has been studied since the early 1990s, primarily through comparisons with the corresponding chromatic polynomial. It is known that for any graph $G$ there is a $k\in \mathbb{N}$ such that $P_{\mathrm{\ell }}(G,m)=P(G,m)$ whenever $m\ge k$. The list color function threshold of $G$, denoted $\tau (G)$, is the smallest $k \geq \chi(G)$suchthat$P_{\ell}(G,m) = P(G,m)$whenever$m \geq k$. In 2009, Thomassen asked whether there is a universal constant $\alpha$ such that for any graph $G$, $\tau (G)\le {\chi }_{\mathrm{\ell }}(G)+\alpha$, where ${\chi }_{\mathrm{\ell }}(G)$ is the list chromatic number of $G$. We show that the answer to this question is no by proving that there exists a constant $C$ such that $\tau(K_{2,l}) - \chi_{\ell}(K_{2,l}) \ge C\sqrt{l}$for$l \ge 16$.