359 research outputs found
The -genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
The obstructions for toroidal graphs with no 's
Forbidden minors and subdivisions for toroidal graphs are numerous. We
consider the toroidal graphs with no -subdivisions that coincide with
the toroidal graphs with no -minors. These graphs admit a unique
decomposition into planar components and have short lists of obstructions. We
provide the complete lists of four forbidden minors and eleven forbidden
subdivisions for the toroidal graphs with no 's and prove that the
lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail
The Z_2-Genus of Kuratowski Minors
A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t x t grid or one of the following so-called t-Kuratowski graphs: K_{3,t}, or t copies of K_5 or K_{3,3} sharing at most 2 common vertices. We show that the Z_2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its Z_2-genus, solving a problem posed by Schaefer and Stefankovic, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces
Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4
We find a graph of genus and its drawing on the orientable surface of
genus with every pair of independent edges crossing an even number of
times. This shows that the strong Hanani-Tutte theorem cannot be extended to
the orientable surface of genus . As a base step in the construction we use
a counterexample to an extension of the unified Hanani-Tutte theorem on the
torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem
Combinatorics of embeddings
We offer the following explanation of the statement of the Kuratowski graph
planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas
intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to
every cell there corresponds a unique cell with the complementary set of
vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere;
there exist precisely two 3-dimensional dichotomial cell complexes, and their
1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their
1-skeleta all but one graphs of the Petersen family.
In higher dimensions n>2, we observe that in order to characterize those
compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited
minors", it suffices to establish finiteness of the list of all (n-1)-connected
n-dimensional finite cell complexes that do not embed in S^{2n} yet all their
proper subcomplexes and proper cell-like combinatorial quotients embed there.
Our main result is that this list contains the n-skeleta of (2n+1)-dimensional
dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell
complexes include (apart from the three joins of the i-skeleta of
(2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on
transforms of dichotomial spheres reworked to include more details;
subsection 2.D "Algorithmic issues" added, etc
Dynamics of Hilbert nonexpansive maps
In his work on the foundations of geometry, Hilbert observed that a formula
which appeared in works by Beltrami, Cayley, and Klein, gives rise to a
complete metric on any bounded convex domain. Some decades later, Garrett
Birkhoff and Hans Samelson noted that this metric has interesting applications,
when considering certain maps of convex cones that contract the metric. Such
situations have since arisen in many contexts, pure and applied, and could be
called nonlinear Perron-Frobenius theory. This note centers around one
dynamical aspect of this theory.Comment: 10 pages. To appear in the Handbook of Hilbert Geometr
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