Coloring graphs with locally few colors

Abstract Let G be a graph, m > r ⩾1 integers. Suppose that it has a good coloring with m colors which uses at most r colors in the neighborhood of every vertex. We investigate these so-called local r -colorings. One of our results (Theorem 2.4) states: The chromatic number of G , Chr( G )⩽ r 2 r log 2 log 2 m (and this value is the best possible in a certain sense). We consider infinite graphs as well.