Enumeratively Chromatic-Choosable Theta Graphs

Chromatic choosability is a notion of fundamental importance in list coloring.

A graph $G$ is chromatic-choosable when its chromatic number, $\chi(G)$, is equal to its list chromatic number $\chi_{\ell}(G)$.

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 $G$, $P(G,m)$.

A graph $G$ is said to be enumeratively chromatic-choosable when $P_{\ell}(G,m)=P(G,m)$ for every $m \in \mathbb{N}$.

Theta graphs and their generalizations have played an important role in graph coloring problems over the years; for example, they appear in the characterization of chromatic-choosable graphs with chromatic number 2.

In this paper we characterize the enumeratively chromatic-choosable theta graphs.

Our proof utilizes ideas from DP-coloring (a.k.a. correspondence coloring), providing yet another example of how the more general setting of DP-coloring can be leveraged to attack a problem in list coloring.