  Proportional choosability is a list coloring analogue of equitable coloring.
Specifically, a $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of
$k$ available colors to each $v \in V(G)$. An $L$-coloring assigns a color to
each vertex $v$ from its list $L(v)$. A proportional $L$-coloring of $G$ is a
proper $L$-coloring in which each color $c \in \bigcup_{v \in V(G)} L(v)$ is
used $\lfloor \eta(c)/k \rfloor$ or $\lceil \eta(c)/k \rceil$ times where
$\eta(c)=\left\lvert{\{v \in V(G) : c \in L(v) \}}\right\rvert$. A graph $G$ is
proportionally $k$-choosable if a proportional $L$-coloring of $G$ exists
whenever $L$ is a $k$-assignment for $G$. Motivated by earlier work, we
initiate the study of proportional choosability with a bounded palette by
studying proportional 2-choosability with a bounded palette. In particular,
when $\ell \geq 2$, a graph $G$ is said to be proportionally $(2,
\ell)$-choosable if a proportional $L$-coloring of $G$ exists whenever $L$ is a
$2$-assignment for $G$ satisfying $|\bigcup_{v \in V(G)} L(v)| \leq \ell$. We
observe that a graph is proportionally $(2,2)$-choosable if and only if it is
equitably 2-colorable. As $\ell$ gets larger, the set of proportionally $(2,
\ell)$-choosable graphs gets smaller. We show that whenever $\ell \geq 5$ a
graph is proportionally $(2, \ell)$-choosable if and only if it is
proportionally 2-choosable. We also completely characterize the connected
proportionally $(2, \ell)$-choosable graphs when $\ell = 3,4$.

Combinatorics