Stars and triangles

A graph is Sm‐full if each vertex belongs to an induced star of order m. The stellarity of a vertex ν in a graph is the maximum size of an induced star centered at ν. Answering a question of Füredi, Mubayi, and West, we show that an Sm‐full finite graph is triangle‐free if the deletion of any edge produces a graph that is not Sm‐full. More generally, a finite graph is triangle‐free if the deletion of any edge lowers the stellarity of at least one of its endpoints. Neither of these results extends to infinite graphs; nevertheless, even an infinite Sm‐full graph has a triangle‐free Sm‐full spanning subgraph. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 170–176, 2001