Circular chromatic indices of even degree regular graphs

Abstract

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 if r∈(2+1k+1,2+1k) for some positive integer k, or r∈(113,4), then there is no graph G with χc′(G)=r. On the other hand, for any odd integer n≥3, if r∈[n,n+14], then there is a simple graph G with χc′(G)=r; if r∈[n,n+13], then there is a multigraph G with χc′(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 χc′(G)=r; if r∈[n,n+1/3], then there is an n-regular multi-graph G with χc′(G)=r.