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On Eggleton and Guy’s conjectured upper bound for the crossing number of the $$n$$-cube. (English) Zbl 0984.05027
Summary: The crossing number $$\nu(G)$$ of a graph $$G$$ is the smallest integer such that there is a drawing for $$G$$ with $$\nu(G)$$ crossings of edges. Let $$Q_n$$ denote the $$n$$-dimensional cube. Eggleton and Guy conjectured in 1970 that $$\nu(Q_n) \leq 4^n\frac {5}{32} - 2^{n-2} \lfloor \frac {n^2+1}{2}\rfloor$$.
We exhibit a drawing for $$n = 6$$ with the same value of Eggleton and Guy’s conjectured upper bound. We construct a family of drawings for the $$n$$@-cubes, $$n \geq 7$$, with number of crossings $$\frac {165}{1024}4^{n}-\frac {2n^2-11n+34}{2}2^{n-2}$$, establishing a new upper bound for $$\nu(Q_n)$$. Our family of drawings confirms Eggleton and Guy’s conjectured upper bound when $$n=7$$ and $$8$$. In addition, our upper bound improves the upper bound $$\nu(Q_n) \leq 4^n\frac {1}{6} - 2^{n-3}n^2 - 2^{n-4}3 + (-2)^n\frac {1}{48}$$ due to Madej.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
topological graph theory; crossing number; drawing
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##### References:
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