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ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS

  • Received : 2010.05.14
  • Published : 2012.05.01

Abstract

In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of $I$-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each $I$-graph $I(n,j,k)$ admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every $I$-graph $I(n,j,k)$ has an isomorphic $I$-graph that admits a unit-distance representation in the Euclidean plane with a $n$-fold rotational symmetry, with the exception of the families $I(n,j,j)$ and $I(12m,m,5m)$, $m{\geq}1$. We also provide unit-distance representations for these graphs.

Keywords

References

  1. B. Alspach, The classification of Hamiltonian generalized Petersen graphs, J. Combin. Theory Ser. B 34 (1983), no. 3, 293-312. https://doi.org/10.1016/0095-8956(83)90042-4
  2. L. W. Berman, A characterization of astral (n4) configurations, Discrete Comput. Geom. 26 (2001), no. 4, 603-612. https://doi.org/10.1007/s00454-001-0041-z
  3. M. Boben and T. Pisanski, Polycyclic configurations, European J. Combin. 24 (2003), no. 4, 431-457. https://doi.org/10.1016/S0195-6698(03)00031-3
  4. M. Boben, T. Pisanski, and A. Zitnik, I-graphs and the corresponding configurations, J. Combin. Des. 13 (2005), no. 6, 406-424.
  5. I. Z. Bouwer, W. W. Chernff, B. Monson, and Z. Star, The Foster Census, Charles Babbage Research Centre, 1988.
  6. F. Buckley and F. Harary, On the Euclidean dimension of a wheel, Graphs Combin. 4 (1988), no. 1, 23-30. https://doi.org/10.1007/BF01864150
  7. K. B. Chilakamarri, The unit-distance graph problem: A brief survey and some new results, Bull. Inst. Combin. Appl. 8 (1993), 39-60.
  8. H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413-455. https://doi.org/10.1090/S0002-9904-1950-09407-5
  9. P. Erdos, F. Harary, and W. T. Tutte, On the dimension of a graph, Mathematika 12 (1965), 118-122. https://doi.org/10.1112/S0025579300005222
  10. R. Frucht, J. E. Graver, and M. E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971), 211-218. https://doi.org/10.1017/S0305004100049811
  11. S. V. Gervacio and I. B. Jos, The Euclidean dimension of the join of two cycles, Discrete Math. 308 (2008), no. 23, 5722-5726. https://doi.org/10.1016/j.disc.2007.10.035
  12. B. Grunbaum, Configurations of Points and Lines, Graduate Studies in Mathematics, 103. American Mathematical Society, Providence, RI, 2009.
  13. H. Hadwiger, Ungeloste probleme no. 40, Elem. Math. 16 (1961), 103-104.
  14. B. Horvat, Unit-distance representations of graphs, Ph.D thesis (in Slovene), University of Ljubljana, 2009.
  15. B. Horvat and T. Pisanski, Unit distance representations of the Petersen graph in the plane, Ars Combin., (to appear).
  16. B. Horvat and T. Pisanski, Products of unit distance graphs, Discrete Math. 310 (2010), no. 12, 1783-1792. https://doi.org/10.1016/j.disc.2009.11.035
  17. B. Horvat, T. Pisanski, and A. Zitnik, Isomorphism checking of I-graphs, Graphs Combin., to appear, doi: 10.1007/s00373-011-1086-2.
  18. M. Lovrecic Sarazin, A note on the generalized Petersen graphs that are also Cayley graphs, J. Combin. Theory Ser. B 69 (1997), no. 2, 226-229. https://doi.org/10.1006/jctb.1997.1729
  19. H. Maehara and V. Rodl, On the dimension to represent a graph by a unit distance graph, Graphs Combin. 6 (1990), no. 4, 365-367. https://doi.org/10.1007/BF01787703
  20. R. Nedela and M. Skoviera, Which generalized Petersen graphs are Cayley graphs?, J. Graph Theory 19 (1995), no. 1, 1-11. https://doi.org/10.1002/jgt.3190190102
  21. T. D. Parsons and T. Pisanski, Vector representations of graphs, Discrete Math. 78 (1989), no. 1-2, 143-154. https://doi.org/10.1016/0012-365X(89)90171-4
  22. M. Petkovsek and H. Zakrajsek, Enumeration of I-graphs: Burnside does it again, Ars Math. Contemp. 2 (2009), no. 2, 241-262.
  23. T. Pisanski and A. Zitnik, Representing Graphs and Maps, Topics in Topological Graph Theory, Series: Encyclopedia of Mathematics and its Applications, No. 129. Cambridge University Press, 2009.
  24. A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators, Springer, illustrated edition, 2008.
  25. A. Steimle and W. Staton, The isomorphism classes of the generalized Petersen graphs, Discrete Math. 309 (2009), no. 1, 231-237. https://doi.org/10.1016/j.disc.2007.12.074
  26. M. E. Watkins, A theorem on Tait colorings with an application to the generalized Petersen graphs, J. Combinatorial Theory 6 (1969), 152-164. https://doi.org/10.1016/S0021-9800(69)80116-X

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