Circular consecutive choosability of graphs

Abstract

This paper considers list circular colouring of graphs in which the colour list assigned to each vertex is an interval of a circle. The circular consecutive choosability chcc(G) of G is defined to be the least t such that for any circle S(r) of length r ≥ χc(G), if each vertex x of G is assigned an interval L(x) of S(r) of length t, then there is a circular r-colouring f of G such that f(x) ∈ L(x). We show that for any finite graph G, χ(G) - 1 ≤ chcc(G) < 2χc(G). We determine the value of chcc(G) for complete graphs, trees, even cycles and balanced complete bipartite graphs. Upper and lower bounds for chcc(G) are given for some other classes of graphs.