A polyhedral investigation of star colorings

Given a weighted undirected graph  GG and a nonnegative integer  kk, the maximum  kk-star colorable subgraph problem consists of finding an induced subgraph of  GG which has maximum weight and can be star colored with at most  kk colors; a star coloring does not color adjacent nodes with the same color and avoids coloring any 4-path with exactly two colors. In this article, we investigate the polyhedral properties of this problem. In particular, we characterize cases in which the inequalities that appear in a natural integer programming formulation define facets. Moreover, we identify graph classes for which these base inequalities give a complete linear description. We then study path graphs in more detail and provide a complete linear description for an alternative polytope for  k=2k=2. Finally, we derive complete balanced bipartite subgraph inequalities and present some computational results.