Vertex coloring without large polychromatic stars

Given an integer k ≥ 2, we consider vertex colorings of graphs in which no k-star subgraph Sk = K1,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 χ̄k⋆(G) and is termed the star-[k] upper chromatic number of G. 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 k) can be given for χ̄k⋆(G), provided that the complement G admits a star-[k]-coloring with more than k colors. © 2011 Elsevier B.V. All rights reserved.