44,424 research outputs found
Removing Even Crossings
An edge in a drawing of a graph is called if it intersects every other edge of the graph an even number of times. Pach and TĂłth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most . We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte
Unexpected behaviour of crossing sequences
The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number
of crossings in a drawing of G on an orientable surface of genus n. We prove
that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) =
b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al.
and resolves a problem of Salazar.Comment: 21 page
The forbidden number of a knot
Every classical or virtual knot is equivalent to the unknot via a sequence of
extended Reidemeister moves and the so-called forbidden moves. The minimum
number of forbidden moves necessary to unknot a given knot is an invariant we
call the {\it forbidden number}. We relate the forbidden number to several
known invariants, and calculate bounds for some classes of virtual knots.Comment: 14 pages, many figures; v2 improves the upper bounds from the
crossing number, and adds more detail to the data presented in the conclusio
Simple realizability of complete abstract topological graphs simplified
An abstract topological graph (briefly an AT-graph) is a pair
where is a graph and is a set of pairs of its edges. The AT-graph is simply
realizable if can be drawn in the plane so that each pair of edges from
crosses exactly once and no other pair crosses. We show that
simply realizable complete AT-graphs are characterized by a finite set of
forbidden AT-subgraphs, each with at most six vertices. This implies a
straightforward polynomial algorithm for testing simple realizability of
complete AT-graphs, which simplifies a previous algorithm by the author. We
also show an analogous result for independent -realizability,
where only the parity of the number of crossings for each pair of independent
edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and
will be included in another pape
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