Blocking nonorientability of a surface

  • Authors:

  • Bojan Mohar;Alexander Schrijver

  • Affiliations:

  • Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia;CWI, Kruislaan 413, Amsterdam, The Netherlands

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

Let S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every one-sided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is "'roughly" equal to one-half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar (Discrete Math. 182 (1998) 245) and replaced by a similar one. In this paper, it is proved that the conjectures in Mohar (1998) and Robertson and Thomas (J. Graph Theory 15 (1991) 407) hold up to a constant error term: For any graph G embedded in S, the orientable genus of G differs from the conjectured value at most by O(g2), where g is the genus of S.