List circular coloring of trees and cycles

Suppose G = (V,E) is a graph and p ≥ 2q are positive integers. A (p,q)-coloring of G is a mapping φ : V → {0,1, . . . ,p − 1} such that for any edge xy of G, q ≤ |φ(x) − φ(y)| ≤ p − q. A color-list is a mapping L : V → P({0,1, . . . ,p − 1}) which assigns to each vertex v a set L(v) of permissible colors. An L-(p,q)-coloring of G is a (p,q)-coloring φ of G such that for each vertex v, φ(v) ∈ L(v). We say G is L-(p,q)-colorable if there exists an L-(p,q)-coloring of G. A color-size-list is a mapping which assigns to each vertex v a non-negative integer (v). We say G is -(p,q)-colorable if for every color-list L with |L(v)| = (v), G is L-(p,q)-colorable. 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 Contract grant sponsor: National Science Council; Contract grant number: NSC942115-M-110-001 Part of the work was done during the stay of X.Z. at LaBRI, supported by the French Ministry of Education. Journal of Graph Theory © 2007 Wiley Periodicals, Inc.