The Erdős and Guy's conjectured equality on the crossing number of hypercubes

Details The Erdős and Guy's conjectured equality on the crossing number of hypercubes The Erdős and Guy's conjectured equality on the crossing number of hypercubes

2012-01-23

arxiv.org/abs/1201.4700v2

Mathematics

Combinatorics

21 pages, 17 figures

Scientific paper

Let $Q_n$ be the $n$-dimensional hypercube, and let ${\rm cr}(Q_n)$ be the \textit{crossing number} of $Q_n$. A long-standing conjecture proposed by Erd\H{o}s and Guy in 1973 is the following equality: ${\rm cr}(Q_n)=(5/32)4^n-\lfloor\frac{n^2+1}{2}\rfloor 2^{n-2}$. In this paper, we construct a drawing of $Q_n$ with less crossings, which implies ${\rm cr}(Q_n)\lneqq (5/32)4^n-\lfloor\frac{n^2+1}{2}\rfloor 2^{n-2}$.

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