Vertex coloring without large polychromatic stars

Given an integer k≥2k≥2, we consider vertex colorings of graphs in which no kk-star subgraph Sk=K1,kSk=K1,k is polychromatic. Equivalently, in a star  -[k][k]-coloring   the closed neighborhood N[v]N[v] of each vertex vv can have at most kk different colors on its vertices. The maximum number of colors that can be used in a star-[k][k]-coloring of graph GG is denoted by χ̄k⋆(G) and is termed the star  -[k][k]upper chromatic number   of GG.We establish some lower and upper bounds on χ̄k⋆(G), and prove an analogue of the Nordhaus–Gaddum theorem. Moreover, a constant upper bound (depending only on kk) can be given for χ̄k⋆(G), provided that the complement G¯ admits a star-[k][k]-coloring with more than kk colors.