We define a random geometric graph by choosing n points in a square and joining pairs of points if they are close to each other. It is natural to ask when standard graph properties (such as connectedness, Hamiltonicity, et cetera) typically occur in this model. We consider the property of containing the square of a Hamilton cycle. Our main result is that for a typical point set the graph contains the square of a Hamilton cycle exactly when a simple local condition is satisfied at every vertex. Perhaps surprisingly, this property exhibits quite different behaviour in the binomial random graph. Furthermore, unlike in the case of connectedness and Hamiltonicity, the local condition is not simply a minimum degree condition.
The emergence of the square of a Hamilton cycle in random geometric graphs
Jack Bartley (QMUL)
Fri, 10/02/2017 - 16:00