Geodesic
New infinite families of almost-planar crossing-critical graphs
The electronic journal of combinatorics, Tome 15 (2008)
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We show that, for all choices of integers $k>2$ and $m$, there are simple $3$-connected $k$-crossing-critical graphs containing more than $m$ vertices of each even degree $\leq2k-2$. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least $7$ in crossing-critical graphs remains open. Furthermore, our newly constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval $\big[3+{1\over5},6-{8\over k+1}\big)$.
DOI : 10.37236/826
Classification : 05C10, 05C62
Mots-clés : crossing critical graphs, almost planar graphs, average degree 
@article{10_37236_826,
     author = {Petr Hlin\v{e}n\'y},
     title = {New infinite families of almost-planar crossing-critical graphs},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/826},
     zbl = {1165.05322},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/826/}
}
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Petr Hliněný. New infinite families of almost-planar crossing-critical graphs. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/826

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