Random graph processes with maximum degree $2$

On the Maximum Degree of a Random Planar Graph

Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(lnn). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable on any fixed surface.

Random Hypergraph Processes with Degree Restrictions

A d-process for s-uniform hypergraphs starts with an empty hypergraph on n vertices, and adds one s-tuple at each time step, chosen uniformly at random from those s-tuples which are not already present as a hyperedge and which consist entirely of vertices with degree less than d. We prove that for d ≥ 2 and s ≥ 3, with probability which tends to 1 as n tends to infinity, the final hypergraph is...

Maximum Likelihood Graph Structure Estimation with Degree Distributions

We describe a generative model for graph edges under specific degree distributions which admits an exact and efficient inference method for recovering the most likely structure. This binary graph structure is obtained by reformulating the inference problem as a generalization of the polynomial time combinatorial optimization problem known as b-matching, which recovers a degree constrained maxim...

The Ramsey number of a graph with bounded maximum degree

The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G. A conjecture of P. Erdos and S. Burr is settled in the afftrmative by proving that for each...

Random Planar Structures and Random Graph Processes