k-Regular graphs with the circular chromatic index close to k

The circular chromatic index of a graph G is the infimum of all rational numbers p/q, such that there exists a circular p/q-edge-coloring of the graph G. It is an interesting problem to determine the set of possible values of the circular chromatic index of k-regular graphs. In this paper, we construct k-regular graphs with circular chromatic index equal to k+a/p for k≥4, p=(2a+1)m+an, a∈{1,2,…,⌊k/2⌋}, and integers m,n≥1, in particular, for all p≥2a2+a+1.