Polynomial-time algorithms for minimum weighted colorings of ($P_5,   \bar{P}_5$)-free graphs and related graph classes
  We design an $O(n^3)$ algorithm to find a minimum weighted coloring of a ($P_5, \bar{P}_5$)-free graph. Furthermore, the same technique can be used to solve the same problem for several classes of graphs, defined by forbidden induced subgraphs, such as (diamond, co-diamond)-free graphs. 
Ch\'inh T. Ho\`ang and D. Adam Lazzarato
cs.DM