Circular chromatic indices of even degree regular graphs

The circular chromatic index   of a graph GG, written χc′(G), is the minimum rr permitting a function c:E(G)→[0,r) such that 1≤|c(e)−c(e′)|≤r−11≤|c(e)−c(e′)|≤r−1 whenever ee and e′e′ are adjacent. It is known that if r∈(2+1k+1,2+1k) for some positive integer kk, or r∈(113,4), then there is no graph GG with χc′(G)=r. On the other hand, for any odd integer n≥3n≥3, if r∈[n,n+14], then there is a simple graph GG with χc′(G)=r; if r∈[n,n+13], then there is a multigraph GG with χc′(G)=r. For most reals rr, it is unknown whether rr is the circular chromatic index of a graph (or a multigraph) or not. In this paper, we prove that for any even integer n≥4n≥4, if r∈[n,n+1/6]r∈[n,n+1/6], then there is an nn-regular simple graph GG with χc′(G)=r; if r∈[n,n+1/3]r∈[n,n+1/3], then there is an nn-regular multi-graph GG with χc′(G)=r.