Abstract
A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.
Article PDF
Similar content being viewed by others
References
Archdeacon, D.: A Kuratowski theorem for the projective plane. J. Graph Theory 5, 243–246 (1981)
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974)
Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Graduate Texts in Mathematics, vol. 115. Springer, New York (1988)
Bredon, G.E.: Topology and Geometry. Graduate Texts in Mathematics, vol. 139. Springer, New York (1997)
Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23, 191–206 (2000)
Cairns, G., McIntyre, M., Nikolayevsky, Y.: The thrackle conjecture for K 5 and K 3,3. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics. Am. Math. Soc., Providence (2004)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (2005)
Dieudonné, J.: A History of Algebraic and Differential Topology. 1900–1960. Birkhäuser, Boston (1989)
Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry, Part III. Springer, Berlin (1990)
Edelen, D.G.B.: Applied Exterior Calculus. Dover, New York (2005)
Giblin, P.J.: Graphs, Surfaces and Homology. Chapman & Hall, Boca Raton (1981)
Gross, J.L., Tucker, T.W.: Topological Graph Theory. Dover, New York (2001)
Grove, L.C.: Classical Groups and Geometric Algebra. Graduate Studies in Mathematics, vol. 39. Am. Math. Soc., Providence (2002)
Lins, S.: Combinatorics of orientation reversing polygons. Aequ. Math. 29, 123–131 (1985)
Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18, 368–376 (1997)
Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Press, Baltimore (2001)
Perlstein, A., Pinchasi, R.: Generalized thrackles and geometric graphs in ℝ3 with no pair of strongly avoiding edges. Preprint
Prasolov, V.V.: Elements of Combinatorial and Differential Topology. Graduate Studies in Mathematics, vol. 74. Am. Math. Soc., Providence (2006)
Woodall, D.R.: Thrackles and deadlock. In: Combinatorial Mathematics and Its Applications, pp. 335–347. Academic Press, San Diego (1971)
Woodall, D.R.: Unsolved problems. In: Combinatorics, Proc. Conf. Combinatorial Math., pp. 359–363. Math. Inst., Oxford (1972)
Zaslavsky, T.: The projective-planar signed graphs. Discrete Math. 113, 223–247 (1993)
Zaslavsky, T.: The order upper bound on parity embedding of a graph. J. Comb. Theory Ser. B 68, 149–160 (1996)
Zieschang, H., Vogt, E., Coldewey, H.-D.: Surfaces and Planar Discontinuous Groups. Lecture Notes in Mathematics, vol. 835. Springer, Berlin (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cairns, G., Nikolayevsky, Y. Generalized Thrackle Drawings of Non-bipartite Graphs. Discrete Comput Geom 41, 119–134 (2009). https://doi.org/10.1007/s00454-008-9095-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-008-9095-5