List circular coloring of trees and cycles

Abstract

Suppose G = (V, E) is a graph and p ≥ 2q are positive integers. A (p, q)-coloring of G is a mapping φ :In this article, we consider list circular coloring of trees and cycles. For any tree T and for any p ≥ 2q, we present a necessary and sufficient condition for T to be -(p, q)-colorable. For each cycle C and for each positive integer k, we present a condition on which is sufficient for C to be -(2k + 1, k)-colorable, and the condition is sharp.

Circular choosability

“…This lemma is closely related to the results of Raspaud and Zhu [17]. Lemma 1 can be considered as a more explicit version of their Lemma 2.5 in which the leaves of the tree are required to have color lists of size one.…”

Circular choosability of graphs

“…In [6], the authors gave a characterization of those for which a tree is -Ô Õµ-choosable, and a characterization of those for which an odd cycle is -´¾ · ½ µ-choosable. The circular chromatic number of a graph has a few equivalent definitions [8].…”

Circular choosability is rational

“…Since its definition by Mohar [4], circular choosability has been studied by several authors including Zhu [17], Havet et al [2], Norine [5], Norine et al [7], Norine and Zhu [8], Raspaud and Zhu [10] and Yu et al [15].…”