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I have a need to generate random planar graphs none of which have an odd cycle, i.e., bipartite graphs. I know there is a substantial two-decade literature on random planar graphs, little with which I …

(1) $\pi(n)$, the number of primes at most $n$, is asymptotic to $n / \ln n$. (2) In the Erdős-Rényi random graph model, $p = \ln n / n$ is a sharp threshold for the connectedness of the graph $G(n,p …

I would like to generate random graphs that might be geometric graphs in some (unknown) dimension. So I would like every triangle in the graph to satisfy the triangle inequality on its (random) edge l …

As far as I know, it is an unsolved question whether or not this is true: If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least double that of its outdeg …

Imagine growing trees from $k$ seeds on a square $n \times n$ region of $\mathbb{Z}^2$. At each step, a unit-length edge $e$ between two points of $\mathbb{Z}^2$ is added. The edge $e$ is chosen rando …