Vertex Colorings without Rainbow or Monochromatic Subgraphs
  This paper investigates vertex colorings of graphs such that some rainbow subgraph~$R$ and some monochromatic subgraph $M$ are forbidden. Previous work focussed on the case that $R=M$. Here we consider the more general case, especially the case that $M=K_2$. 
Wayne Goddard and Honghai Xu
math.CO