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A Solution to Ringel's Circle Problem

James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, Bartosz Walczak, Xavier Goaoc, Michael Kerber
2022
This provides a strong negative answer to Ringel's circle problem (1959).  ...  We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number.  ...  S o C G 2 0 2 2 33:4 A Solution to Ringel's Circle Problem ▶ Gallai's Theorem.  ... 
doi:10.4230/lipics.socg.2022.33 fatcat:4kenmy5j6zasxcj2hrfij2tpsy

Optimal Curve Straightening is ∃R-Complete [article]

Jeff Erickson
2019 arXiv   pre-print
a polygon with m vertices that is isotopic to γ?  ...  Second, for any semi-algebraic set V, there is an integer m and a closed curve γ such that the space of all m-gons isotopic to γ is homotopy equivalent to V.  ...  theory of the reals-the problem of deciding whether a given system of multivariate polynomial equalities and inequalities has a real solution.  ... 
arXiv:1908.09400v2 fatcat:yfz5c3qs4rhyfa3pmig43hgfhu

Embeddings of graphs

Carsten Thomassen
1994 Discrete Mathematics  
A simple answer to Ringel's question (which now seems hopeless in view of Theorem 5.2) might give a unified and simpler solution of the Conjecture, since a graph G triangulates an orientable surface if  ...  This problem was in [20] reduced to the graph genus problem.  ... 
doi:10.1016/0012-365x(92)00062-v fatcat:2dzynzlujfemteeuqtjlexdzhe

On the covering type of a space [article]

Max Karoubi, Charles Weibel
2016 arXiv   pre-print
We introduce the notion of the "covering type" of a space, which is more subtle that the notion of Lusternik Schnirelman category.  ...  It measures the complexity of a space which arises from coverings by contractible subspaces whose non-empty intersections are also contractible.  ...  The authors would like to thank Mike Saks and Patrick Devlin for discussions about color mapping theorems, Ieke Moerdijk for discussions about Quillen's Theorem A, and the referee for numerous expositional  ... 
arXiv:1612.00532v1 fatcat:wmv2vqcdirf35lr6cpuyx73c74

Book announcements

1990 Discrete Applied Mathematics  
Iterative algorithms-for definitions with a finite limit. A path interpretation of minimum solution for a class of recursive definitions.  ...  Applications to group actions on the surface Sz). Current graphs (Ringel's generating rows for Heffter's schemes. Gustin's combinatorial current graphs. Orientable topological current graphs.  ... 
doi:10.1016/0166-218x(90)90074-m fatcat:uhwd4fl3wvcetkvers2zklsbry

Oriented matroids and complete-graph embeddings on surfaces

Jürgen Bokowski, Tomaž Pisanski
2007 Journal of combinatorial theory. Series A  
We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface.  ...  Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces.  ...  An independent proof of the same result, using co-homology again, was given to us by Francisco Santos.  ... 
doi:10.1016/j.jcta.2006.06.012 fatcat:54pjpzxihfgs7kvonzluejtk5a

Solution of the heawood map-coloring problem—Cases 3, 5, 6, and 9

J.W.T. Youngs
1970 Journal of Combinatorial Theory  
This paper gives a proof of the fact that the chromatic number of an orientable surface of genus p is equal to the integral part of (7 + ~/1 + 48p)/2 whenever the latter is congruent to 3, 5, 6, or 9 modulo  ...  Thus the array is a scheme for KI~ (see [10]).  ...  These are unrelated problems in the sense that a solution of one does not depend upon a solution to the other.  ... 
doi:10.1016/s0021-9800(70)80075-8 fatcat:l3hkhqzmlbcthasgfnf4imbtzm

Surface embeddings of Steiner triple systems

M. J. Grannell, T. S. Griggs, Jozef S?ir�n?
1998 Journal of combinatorial designs (Print)  
A Steiner triple system of order n (STS(n)) is said to be embeddable in an orientable surface if there is an orientable embedding of the complete graph K n whose faces can be properly 2-coloured (say,  ...  black and white) in such a way that all black faces are triangles and these are precisely the blocks of the STS(n).  ...  In addition, the authors would like to thank the referees for several helpful comments and for drawing our attention to [20] .  ... 
doi:10.1002/(sici)1520-6610(1998)6:5<325::aid-jcd2>3.0.co;2-h fatcat:txdwnf5vezfqfjasannuh3xbzy

Partitions of complete geometric graphs into plane trees

Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood
2006 Computational geometry  
Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees.  ...  Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning.  ...  Thanks to Susana López and Anna Lladó for bringing this family of problems to our attention.  ... 
doi:10.1016/j.comgeo.2005.08.006 fatcat:3ukn4ikaazfuhjgq2c72p5qh54

Partitions of Complete Geometric Graphs into Plane Trees [chapter]

Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood
2005 Lecture Notes in Computer Science  
Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees.  ...  Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning.  ...  Thanks to Susana López and Anna Lladó for bringing this family of problems to our attention.  ... 
doi:10.1007/978-3-540-31843-9_9 fatcat:nekcgxvzljaw5jf6plc2bigxhq

SUICIDAL PATIENTS

Jan Prasko, Tomas Diveky, Ales Grambal, Klara Latalova
2010 Biomedical Papers of the Faculty of Medicine of Palacky University  
Patient protection, usually in the setting of a closed psychiatric ward, is necessary if he or she has a clear plan and means to commit suicide.  ...  Suicide attempts are a frequent cause of hospital admissions, in particular to anesthesiology and resuscitation departments. Risk factors. Women attempt suicide three times more often than men.  ...  The act of suicide may be put off to give a chance to alternative solutions (Table 9 ).  ... 
doi:10.5507/bp.2010.041 pmid:21048815 fatcat:mectrcqw4jbqjbqposq2m6ejjq

Interview with Tomaž Pisanski

ECA, University of Haifa
2021 Enumerative Combinatorics and Applications  
Tomaž Pisanski studied at the University of Ljubljana where he obtained a B.Sc., M.Sc., and Ph.D. in mathematics. He completed his Ph.D. thesis under the guidance of Torrence Parsons in 1981.  ...  Currently, Pisanski is a professor of discrete and computational mathematics and head of the Department of Information Sciences and Technology at the University of Primorska in Koper.  ...  However, if the series is new, one is faced with a natural and sometimes challenging problem of finding a closed-form solution for it.  ... 
doi:10.54550/eca2022v2s3i11 fatcat:d44icay3yjflti3ejjnrq6og2i

Monotone Simultaneous Embeddings of Upward Planar Digraphs

Oswin Aichholzer, Thomas Hackl, Sarah Lutteropp, Tamara Mchedlidze, Alexander Pilz, Birgit Vogtenhuber
2015 Journal of Graph Algorithms and Applications  
with respect to the given directions, including the construction of a solution if it exists.  ...  Finally, we discuss complexity issues related to our problems.  ...  ., Ringel's construction using Pappus' Theorem).  ... 
doi:10.7155/jgaa.00350 fatcat:46tdmqjuovdj7i2gzg3qucpdce

Monotone Simultaneous Embedding of Directed Paths [article]

Oswin Aichholzer and Thomas Hackl and Sarah Lutteropp and Tamara Mchedlidze and Alexander Pilz and Birgit Vogtenhuber
2014 arXiv   pre-print
with respect to the given directions, including the construction of a solution if it exists.  ...  Finally, we discuss complexity issues related to our problems.  ...  Research on this topic was initiated during a research visit of Sarah Lutteropp and Tamara Mchedlidze in February 2013 in Graz, Austria.  ... 
arXiv:1310.6955v2 fatcat:dpvxxprvpfhnrlepbnpzczmur4

Arrangements of Approaching Pseudo-Lines [article]

Stefan Felsner, Alexander Pilz, Patrick Schnider
2020 arXiv   pre-print
type contains a bichromatic triangular cell.  ...  We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements  ...  , at which they cross, and 3. there is a bijection L from the unit circle C to the family of curves such that L(p) is a continuous function (under the Hausdorff metric) of p ∈ C.  ... 
arXiv:2001.08419v1 fatcat:fdddu325gzhkbjjpofaanxbubm
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