Circular chromatic indices of even degree regular graphs

Abstract

a b s t r a c tThe circular chromatic index of a graph G, written χ ′ c (G), is the minimum r permitting a function c :On the other hand, for any odd integer n ≥ 3, if r ∈ [n, n + 1 4 ], then there is a simple graph G with χ ′ c (G) = r; if r ∈ [n, n + 1 3 ], then there is a multigraph G with χ ′ c (G) = r. For most reals r, it is unknown whether r is the circular chromatic index of a graph (or a multigraph) or not. In this paper, we prove that for any even integer n ≥ 4, if r ∈ [n, n + 1/6], then there is an n-regular simple graph G with χ ′ c (G) = r; if r ∈ [n, n + 1/3], then there is an n-regular multi-graph G with χ ′ c (G) = r.