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Let G(v,e) be the set of all simple graphs with v vertices and e edges and let P2(G) = P d 2 denote the sum of the squares of the degrees, d1,...,dv, of the vertices of G. It is known that the… (More)

Given integers $k\geq1$ and $n\geq0$, there is a unique way of writing $n$ as $n=\binom{n_{k}}{k}+\binom{n_{k-1}}{k-1}+...+\binom{n_{1}}{1}$ so that $0\leq n_{1}<...<n_{k-1}<n_{k}$. Using this… (More)

Even the most superficial glance at the vast majority of crossing-minimal geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or… (More)

- Bernardo M. 'Abrego, Oswin Aichholzer, +6 authors Birgit Vogtenhuber
- 2015

The Harary-Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is \[ H(n) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut… (More)

Let Q be a finite set of points in the plane. For any set P of points in the plane, SQ(P) denotes the number of similar copies of Q contained in P. For a fixed n, Erdýos and Purdy asked to determine… (More)

A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k… (More)

For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$… (More)