Anusch Taraz

On random planar graphs, the number of planar graphs and their triangulations

Let Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that |Pn| ≤ n! (75.8)n+o(n) and Bender, Gao and Wormald proved that |Pn| ≥ n! (26.1)n+o(n). Gerke and McDiarmid proved that almost all graphs in Pn have at least 13/7n edges. In this paper, we show that |Pn| ≤ n! (37.3)n+o(n) and that almost all graphs in Pn have at most 2.56n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least e3(3n-m)/2 labelled triangulations on n vertices, where e is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible.

Large planar subgraphs in dense graphs

We prove sufficient and essentially necessary conditions in terms of the minimum degree for a graph to contain planar subgraphs with many edges. For example, for all positive γ every sufficiently large graph G with minimum degree at least (2/3+γ)|G| contains a triangulation as a spanning subgraph, whereas this need not be the case when the minimum degree is less than 2|G|/3.

For Which Densities are Random Triangle-Free Graphs Almost Surely Bipartite?

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Hypergraph Packing and Graph Embedding

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Asymptotic enumeration, global structure, and constrained evolution

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K 4 -free subgraphs of random graphs revisited

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