Some recent results on niche graphs

Abstract In an earlier paper entitled “Niche graphs” written by Cable, Jones, Lundgren and Seager, niche graphs were introduced and examples were provided of graphs which have niche number 0, 1, 2, and ∞. However, no examples were found of a niche graph having finite niche number 3 or larger. We still have had no success in our efforts to find such a graph. Nevertheless we have gotten some interesting results. For example, we show in this paper that if there is such a graph, then there must be one which is connected. We also show that the niche number of a graph which has a finite niche number is ≤ 2 3 |V(G)|. In addition we determine the niche number of all “wheel” graphs.