k-Regular graphs with the circular chromatic index close to

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 = ( 2 a + 1 ) m + a n , a ∈ { 1 , 2 , … , ⌊ k / 2 ⌋ } , and integers m , n ≥ 1 , in particular, for all p ≥ 2 a 2 + a + 1 .