  For a multigraph G, the integer round-up phi(G) of the fractional chromatic
index yields a good general lower bound for the chromatic index . For an upper
bound, Kahn showed that for any real c > 0 there exists a positive integer N so
that the chromatic index is less than (1+c)*phi(G) whenever the fractional
index > N. We show the amount by which the chromatic index can surpass phi(G)
is in fact logarithmic, by showing that for any multigraph G with order n > 3
and at least one edge, the chromatic index is less than phi(G) + log (min
{(n+1)/3, phi(G)}) .
