A sublinear bound on the chromatic index of multigraphs

The integer round-up 4(G) of the fractional chromatic index yields the standard lower bound for the chromatic index of a multigraph G. We show that if G has even order n, then the chromatic index exceeds 4(G) by at most max{log,,, n, 1 + n/30}. More generally, we show that for any real b, 2/3 O, there exists a positive integer N such that x(G) 4(G) < cn for any multigraph G with order n > N. @ 1999 Elsevier Science B.V. All rights reserved