Vertex coloring without large polychromatic stars

Given an integer k ≥ 2 , we consider vertex colorings of graphs in which no k -star subgraph S k = K 1 , k is polychromatic. Equivalently, in a star- [ k ] -coloring the closed neighborhood N [ v ] of each vertex v can have at most k different colors on its vertices. The maximum number of colors that can be used in a star- [ k ] -coloring of graph G is denoted by χ ̄ k ⋆ ( G ) and is termed the star- [ k ] upper chromatic number of G . ablish some lower and upper bounds on χ ̄ k ⋆ ( G ) , and prove an analogue of the Nordhaus–Gaddum theorem. Moreover, a constant upper bound (depending only on k ) can be given for χ ̄ k ⋆ ( G ) , provided that the complement G ¯ admits a star- [ k ] -coloring with more than k colors.