Some recent results on niche graphs

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. The authors still have had no success in their efforts to find such a graph. Nevertheless they have gotten some interesting results. For example, the authors show in this paper that if there is such a graph, then there must be one which is connected. They also show that the niche number of a graph which has a finite niche number is . In addition the authors determine the niche number of all ‘wheel’ graphs.