  In 1999, McDiarmid and Reed conjectued that the approximation ratio $9/8$ of
multichromatic number to weighted clique number asymptotically is the best
possible for general weighted hexagonal graphs. We prove that there is a proper
multicoloring of $G$ that uses at most $15 \lfloor\frac{\omega(G)}{12}\rfloor+
18 $ colors improving the best previously known asimptotic ratio from 4/3 to
5/4.
