Strong Hanani-Tutte for the Torus

If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus

Strong Hanani-Tutte on the Projective Plane

If a graph can be drawn in the projective plane so that every two non-adjacent edges cross an even number of times, then the graph can be embedded in the projective plane

Dominating Sets in Plane Triangulations

In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.Comment: 14 pages, 6 figures; Revised lemmas 6-8, clarified arguments and fixed typos, result unchange

Statistical models for cores decomposition of an undirected random graph

The $k$-core decomposition is a widely studied summary statistic that describes a graph's global connectivity structure. In this paper, we move beyond using $k$-core decomposition as a tool to summarize a graph and propose using $k$-core decomposition as a tool to model random graphs. We propose using the shell distribution vector, a way of summarizing the decomposition, as a sufficient statistic for a family of exponential random graph models. We study the properties and behavior of the model family, implement a Markov chain Monte Carlo algorithm for simulating graphs from the model, implement a direct sampler from the set of graphs with a given shell distribution, and explore the sampling distributions of some of the commonly used complementary statistics as good candidates for heuristic model fitting. These algorithms provide first fundamental steps necessary for solving the following problems: parameter estimation in this ERGM, extending the model to its Bayesian relative, and developing a rigorous methodology for testing goodness of fit of the model and model selection. The methods are applied to a synthetic network as well as the well-known Sampson monks dataset.Comment: Subsection 3.1 is new: `Sample space restriction and degeneracy of real-world networks'. Several clarifying comments have been added. Discussion now mentions 2 additional specific open problems. Bibliography updated. 25 pages (including appendix), ~10 figure