Enumerative Chromatic Choosability

Chromatic-choosability is a notion of fundamental importance in list coloring. A graph is chromatic-choosable when its chromatic number is equal to its list chromatic number. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted $P_{\ell}(G,m)$, which is the list analogue of the chromatic polynomial of $G$, $P(G,m)$. It is known that for any graph $G$ there is a positive integer $k$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. In this paper, we study enumerative chromatic-choosability. A graph $G$ is enumeratively chromatic-choosable when $P_{\ell}(G,m) = P(G,m)$ whenever $m \in \mathbb{N}$. We completely determine the graphs of chromatic number two that are enumeratively chromatic-choosable. We construct examples of graphs that are chromatic-choosable but fail to be enumeratively chromatic-choosable, and finally, we explore a conjecture as to whether for every graph $G$, there is a $p \in \mathbb{N}$ such that the join of $G$ and $K_p$ is enumeratively chromatic-choosable. The techniques we use to prove results are diverse and include probabilistic ideas and ideas from DP (or correspondence)-coloring.