The crossing number of the generalized Petersen graph P [ 3 k , k ]

Stanley Fiorini; John Baptist Gauci

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 4, page 337-347
  • ISSN: 0862-7959

Abstract

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Guy and Harary (1967) have shown that, for k 3 , the graph P [ 2 k , k ] is homeomorphic to the Möbius ladder M 2 k , so that its crossing number is one; it is well known that P [ 2 k , 2 ] is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of P [ 2 k + 1 , 2 ] is three, for k 2 . Fiorini (1986) and Richter and Salazar (2002) have shown that P [ 9 , 3 ] has crossing number two and that P [ 3 k , 3 ] has crossing number k , provided k 4 . We extend this result by showing that P [ 3 k , k ] also has crossing number k for all k 4 .

How to cite

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Fiorini, Stanley, and Gauci, John Baptist. "The crossing number of the generalized Petersen graph $P[3k,k]$." Mathematica Bohemica 128.4 (2003): 337-347. <http://eudml.org/doc/249227>.

@article{Fiorini2003,
abstract = {Guy and Harary (1967) have shown that, for $k\ge 3$, the graph $P[2k,k]$ is homeomorphic to the Möbius ladder $\{M_\{2k\}\}$, so that its crossing number is one; it is well known that $P[2k,2]$ is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of $P[2k+1,2]$ is three, for $k\ge 2.$ Fiorini (1986) and Richter and Salazar (2002) have shown that $P[9,3]$ has crossing number two and that $P[3k,3]$ has crossing number $k$, provided $k\ge 4$. We extend this result by showing that $P[3k,k]$ also has crossing number $k$ for all $k\ge 4$.},
author = {Fiorini, Stanley, Gauci, John Baptist},
journal = {Mathematica Bohemica},
keywords = {graph; drawing; crossing number; generalized Petersen graph; Cartesian product; graph; drawing; crossing number; generalized Petersen graph; Cartesian product},
language = {eng},
number = {4},
pages = {337-347},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The crossing number of the generalized Petersen graph $P[3k,k]$},
url = {http://eudml.org/doc/249227},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Fiorini, Stanley
AU - Gauci, John Baptist
TI - The crossing number of the generalized Petersen graph $P[3k,k]$
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 4
SP - 337
EP - 347
AB - Guy and Harary (1967) have shown that, for $k\ge 3$, the graph $P[2k,k]$ is homeomorphic to the Möbius ladder ${M_{2k}}$, so that its crossing number is one; it is well known that $P[2k,2]$ is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of $P[2k+1,2]$ is three, for $k\ge 2.$ Fiorini (1986) and Richter and Salazar (2002) have shown that $P[9,3]$ has crossing number two and that $P[3k,3]$ has crossing number $k$, provided $k\ge 4$. We extend this result by showing that $P[3k,k]$ also has crossing number $k$ for all $k\ge 4$.
LA - eng
KW - graph; drawing; crossing number; generalized Petersen graph; Cartesian product; graph; drawing; crossing number; generalized Petersen graph; Cartesian product
UR - http://eudml.org/doc/249227
ER -

References

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  1. 10.7146/math.scand.a-11910, Math. Scand. 48 (1981), 184–188. (1981) MR0631334DOI10.7146/math.scand.a-11910
  2. On the crossing number of generalized Petersen graphs, Ann. Discrete Math. 30 (1986), 225–242. (1986) Zbl0595.05030MR0861299
  3. 10.4153/CMB-1967-046-4, Canad. Math. Bull. 10 (1967), 493–496. (1967) MR0224499DOI10.4153/CMB-1967-046-4
  4. On the crossing numbers of S m × C n , Čas. Pěst. Mat. 107 (1982), 225–230. (1982) MR0673046
  5. 10.4064/fm-15-1-271-283, Fund. Math. 15 (1930), 271–283. (1930) DOI10.4064/fm-15-1-271-283
  6. 10.1007/s003730200028, Graphs Combin. 18 (2002), 381–394. (2002) MR1913677DOI10.1007/s003730200028

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