The circular chromatic index of a graph G, written χc′(G), is the minimum r permitting a function c:E(G)→[0,r) such that 1≤c(e)−c(e′)≤r−1 whenever e and e′ are adjacent. It is known that for any ε∈(0,1/3), there is a 3‐regular simple graph G with χc′(G)=3+ε. This article proves the following results: Assume n≥5 is an odd integer. For any ε∈(0,1/4), there is an n‐regular simple graph G with χc′(G)=n+ε. For any ε∈(0,1/3), there is an n‐regular multigraph G with χc′(G)=n+ε.

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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.