Abstract:
 Let $S(r)$ denote a circle of circumference $r$. The circular
consecutive choosability $ch_{cc}(G)$ of a graph $G$ is the least
real number $t$ such that for any $r \geq \chi_c(G)$, if each vertex
$v$ is assigned a closed interval $L(v)$ of length $t$ on $S(r)$,
then there is a circular $r$colouring $f$ of $G$ such that $f(v)
\in L(v)$. We investigate, for a graph, the relations between its
circular consecutive choosability and choosibility. It is proved
that for any positive integer $k$, if a graph $G$ is $k$choosable,
then $ch_{cc}(G) \le k+11/k$; moreover, the bound is sharp for $k
\ge 3$. For $k=2$, it is proved that if $G$ is $2$choosable then
$ch_{cc}(G) \le 2$, while the equality holds if and only if $G$
contains a cycle. In addition, we prove that there exist circular
consecutive $2$choosable graphs which are not $2$choosable. In
particular, it is shown that $ch_{cc}(G)=2$ holds for all cycles and
for $K_{2,n}$ with $n \geq 2$. On the other hand, we prove that $ch_{cc}(G) > 2$ holds for many generalized theta graphs.
