The smallest graphs with certain adjacency properties

Abstract

AbstractA graph is said to have property P1,n if for every sequence of n + 1 points, there is another point adjacent only to the first point. It has previously been shown that almost all graphs have property P1,n. It is easy to verify that for each n, there is a cube with this property. A more delicate question asks for the construction of the smallest graphs having property P1,n. We find that this problem is intimately related with the discovery of the highly symmetric graphs known as cages, and are thereby enabled to resolve this question for 1⩽n⩽6