# MPI-I-91-124

## On crossing numbers of hypercubes and cube connected cycles

### Sýkora, Ondrej and Vrto, Imrich

#### November 1991, 6 pages.

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##### Status: available - back from printing

Recently the hypercube-like networks have received
considerable attention in the field of parallel computing due to its
high potential for system availability and parallel execution of
algorithms.
The crossing number ${\rm cr}(G)$ of a graph $G$ is
defined as the least
number of crossings of its edges when $G$ is drawn in a plane.
Crossing numbers naturally appear in the fabrication of VLSI circuit
and provide a good
area lower bound argument in VLSI complexity theory.
According to the survey paper of Harary et al.,
all that is known on the exact
values of an n-dimensional hypercube
${\rm cr}(Q_n)$ is ${\rm cr}(Q_3)=0, {\rm cr}(Q_4)=8$ and
${\rm cr}(Q_5)\le 56.$
We prove the following tight bounds on ${\rm cr}(Q_n)$ and
${\rm cr}(CCC_n)$:
\[ \frac{4^n}{20} - (n+1)2^{n-2} < {\rm cr}(Q_n) < \frac{4^n}{6}
-n^22^{n-3} \]
\[ \frac{4^n}{20} - 3(n+1)2^{n-2} < {\rm cr}(CCC_n) < \frac{4^n}{6} +
3n^22^{n-3}. \]
Our lower bounds
on ${\rm cr}(Q_n)$ and ${\rm cr}(CCC_n)$ give immediately
alternative proofs that the area complexity of
{\it hypercube} and $CCC$
computers realized on VLSI circuits is $A=\Omega (4^n)$

**URL to this document: **https://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1991-124

**BibTeX**
`@TECHREPORT{``SykoraVrto91a``,`

` AUTHOR = {Sýkora, Ondrej and Vrto, Imrich},`

` TITLE = {On crossing numbers of hypercubes and cube connected cycles},`

` TYPE = {Research Report},`

` INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-91-124},`

` MONTH = {November},`

` YEAR = {1991},`

` ISSN = {0946-011X},`

`}`