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Generalized thrackle drawings of non-bipartite graphs. (English) Zbl 1191.05032
Summary: 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 \(\chi (M) - 1\). As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C62 Graph representations (geometric and intersection representations, etc.)
68R10 Graph theory (including graph drawing) in computer science
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[1] Archdeacon, D., A Kuratowski theorem for the projective plane, J. Graph Theory, 5, 243-246, (1981) · Zbl 0464.05028
[2] Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974) · Zbl 0284.05101
[3] Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Graduate Texts in Mathematics, vol. 115. Springer, New York (1988) · Zbl 0629.53001
[4] Bredon, G.E.: Topology and Geometry. Graduate Texts in Mathematics, vol. 139. Springer, New York (1997) · Zbl 0934.55001
[5] Cairns, G.; Nikolayevsky, Y., Bounds for generalized thrackles, Discrete Comput. Geom., 23, 191-206, (2000) · Zbl 0959.05030
[6] Cairns, G.; McIntyre, M.; Nikolayevsky, Y.; Pach, J. (ed.), The thrackle conjecture for \(K\_{}\{5\}\) and \(K\_{}\{3,3\}, (2004),\) Providence · Zbl 1061.05026
[7] Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (2005) · Zbl 1074.05001
[8] Dieudonné, J.: A History of Algebraic and Differential Topology. 1900-1960. Birkhäuser, Boston (1989) · Zbl 0673.55002
[9] Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry, Part III. Springer, Berlin (1990) · Zbl 0703.55001
[10] Edelen, D.G.B.: Applied Exterior Calculus. Dover, New York (2005) · Zbl 1099.58001
[11] Giblin, P.J.: Graphs, Surfaces and Homology. Chapman & Hall, Boca Raton (1981) · Zbl 0477.57001
[12] Gross, J.L., Tucker, T.W.: Topological Graph Theory. Dover, New York (2001) · Zbl 0991.05001
[13] Grove, L.C.: Classical Groups and Geometric Algebra. Graduate Studies in Mathematics, vol. 39. Am. Math. Soc., Providence (2002)
[14] Lins, S., Combinatorics of orientation reversing polygons, Aequ. Math., 29, 123-131, (1985) · Zbl 0592.05019
[15] Lovász, L.; Pach, J.; Szegedy, M., On Conway’s thrackle conjecture, Discrete Comput. Geom., 18, 368-376, (1997) · Zbl 0892.05017
[16] Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Press, Baltimore (2001) · Zbl 0979.05002
[17] Perlstein, A., Pinchasi, R.: Generalized thrackles and geometric graphs in ℝ3 with no pair of strongly avoiding edges. Preprint · Zbl 1188.05055
[18] Prasolov, V.V.: Elements of Combinatorial and Differential Topology. Graduate Studies in Mathematics, vol. 74. Am. Math. Soc., Providence (2006) · Zbl 1098.57001
[19] Woodall, D. R., Thrackles and deadlock, 335-347, (1971), San Diego · Zbl 0213.50603
[20] Woodall, D. R., Unsolved problems, 359-363, (1972), Oxford
[21] Zaslavsky, T., The projective-planar signed graphs, Discrete Math., 113, 223-247, (1993) · Zbl 0779.05018
[22] Zaslavsky, T., The order upper bound on parity embedding of a graph, J. Comb. Theory Ser. B, 68, 149-160, (1996) · Zbl 0856.05030
[23] Zieschang, H., Vogt, E., Coldewey, H.-D.: Surfaces and Planar Discontinuous Groups. Lecture Notes in Mathematics, vol. 835. Springer, Berlin (1980) · Zbl 0438.57001
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