Circular consecutive choosability of k-choosable graphs

Let S(r) denote a circle of circumference r. The circular consecutive choosability chcc(G) of a graph G is the least real number t such that for any r > χc(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) ∈ L(v). We investigate, for a graph, the relations between its circular consecutive choosability and choosability. It is proved that for any positive integer k, if a graph G is k-choosable, then chcc(G) 6 k + 1− 1/k; moreover, the bound is sharp for k > 3. For k = 2, it is proved that if G is 2-choosable then chcc(G) 6 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 2choosable. In particular, it is shown that chcc(G) = 2 holds for all cycles and for K2,n with n > 2. On the other hand, we prove that chcc(G) > 2 holds for many generalized theta graphs.