Two maps with large representativity on one surface Article uri icon

abstract

  • We show that, for each orientable surface Σ, there is a constant cΣ so that, if G1 and G2 are embedded simultaneously in Σ, with representativities r1 and r 2, respectively, then the minimum number cr(G1, G 2 of crossings between the two maps satisfies cr(G1,G 2) ≤cΣ/r1r2 |E(G1) ||E(G2)|. This refines earlier estimates by Negami. Furthermore, we provide a counterexample to a conjecture of Archdeacon and Bonnington by exhibiting, for each k, embeddings G1 and G2 in the double torus so that, if we force all the vertices of G1 to be in the same face of G2 then the number of crossings between G1 and G2 is at least k. cr(G1, G2). © 2005 Wiley Periodicals, Inc.

publication date

  • 2005-01-01