Results 1 
6 of
6
Removing even crossings
 J. COMBINAT. THEORY, SER. B
, 2005
"... An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth 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 ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth 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 (the only proof we know of not to use 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 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.
Removing even crossings on surfaces
 EUROPEAN JOURNAL OF COMBINATORICS, 30(7):1704
, 2009
"... We give a new, topological proof that the weak HananiTutte theorem is true on orientable surfaces and extend the result to nonorientable surfaces. That is, we show that if a graph G cannot be embedded on a surface S, then any drawing of G on S must contain two edges that cross an odd number of time ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
We give a new, topological proof that the weak HananiTutte theorem is true on orientable surfaces and extend the result to nonorientable surfaces. That is, we show that if a graph G cannot be embedded on a surface S, then any drawing of G on S must contain two edges that cross an odd number of times. We apply the result and proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle in a surface S can be embedded in S. We also extend to arbitrary surfaces a result of Pach and Tóth that allows the redrawing of a graph so as to remove all crossings with even edges (an edge is even if it crosses every other edge an even number of times). From this result we can conclude that crS(G), the crossing number of a graph G on surface S, is bounded by 2 ocrS(G) 2, where ocr S(G) is the odd crossing number of G on surface S. Finally, we show that ocrS(G) =crS(G) whenever ocrS(G) ≤ 2, for any surface S.
HananiTutte and Related Results
, 2011
"... We are taking the view that crossings of adjacent edges are trivial, and easily got rid of. ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
We are taking the view that crossings of adjacent edges are trivial, and easily got rid of.
Removing even crossings, continued
, 2006
"... In this paper we investigate how certain results related to the HananiTutte theorem can be lifted to orientable surfaces of higher genus. We give a new simple, geometric proof that the weak HananiTutte theorem is true for highergenus surfaces. We extend the proof to prove that bipartite generaliz ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper we investigate how certain results related to the HananiTutte theorem can be lifted to orientable surfaces of higher genus. We give a new simple, geometric proof that the weak HananiTutte theorem is true for highergenus surfaces. We extend the proof to prove that bipartite generalized thrackles in a surface S can be embedded in S. We also show that a result of Pach and Tóth that allows the redrawing of a graph removing intersections on even edges remains true on highergenus surfaces. As a consequence, we can conclude that crS(G), the crossing number of the graph G on surface S, is bounded by 2 ocrS(G) 2, where ocr(G)S is the odd crossing number of G on surface S. Finally, we begin an investigation of optimal crossing configurations for which ocr ∼ = cr.
EMBEDDINGS OF PFAFFIAN BRACES AND POLYHEX GRAPHS
, 2009
"... Let G be a graph admitting a perfect matching. A cycle of even size C is central if G − C has a perfect matching. Given an orientation to G, an even cycle C is oddly oriented if along either direction of traversal around C, the number of edges of C with the direction as the same as the traversal di ..."
Abstract
 Add to MetaCart
(Show Context)
Let G be a graph admitting a perfect matching. A cycle of even size C is central if G − C has a perfect matching. Given an orientation to G, an even cycle C is oddly oriented if along either direction of traversal around C, the number of edges of C with the direction as the same as the traversal direction is odd. An orientation of G is Pfaffian if every central cycle of G is oddly oriented. A graph G is Pfaffian if it has a Pfaffian orientation. In this paper, we show that every embedding of a Pfaffian brace on a surface with positive genus has facewidth at most three and that the cyclic edgeconnectivity of a Pfaffian cubic brace different from the Heawood graph is four. Finally, we characterize all Pfaffian polyhex graphs.