Improved approximative multicoloring of hexagonal graphs

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.