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 {\em circular consecutive choosability} $ch_{cc}(G)$ of $G$ is defined to be the least $t$ such that for any circle $S(r)$ of length $r \geq \chi_c(G)$, 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 $f(x) \in L(x)$. We show that for any finite graph $G$, $\chi(G)-1 \leq ch_{cc}(G) \lt 2 \chi_c(G)$. We determine the value of $ch_{cc}(G)$ for complete graphs, trees, even cycles and balanced complete bipartite graphs. Upper and lower bounds for $ch_{cc}(G)$ are given for some other classes of graphs.