20 Years of Negami’s Planar Cover Conjecture

@article{Hlinn201020YO,
  title={20 Years of Negami’s Planar Cover Conjecture},
  author={Petr Hliněn{\'y}},
  journal={Graphs and Combinatorics},
  year={2010},
  volume={26},
  pages={525-536}
}
  • P. Hliněný
  • Published 2010
  • Mathematics, Computer Science
  • Graphs and Combinatorics
In 1988, Seiya Negami published a conjecture stating that a graph G has a finite planar cover (i.e. a homomorphism from some planar graph onto G which maps the vertex neighbourhoods bijectively) if and only if G embeds in the projective plane. Though the “if” direction is easy, and over ten related research papers have been published during the past 20 years of investigation, this beautiful conjecture is still open in 2008. We give a short accessible survey on Negami’s conjecture and all the… Expand

Figures and Topics from this paper

How Not to Characterize Planar-Emulable Graphs
We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph)—a problem raised in Fellows' thesis (1985) and conceptually relatedExpand
How not to characterize planar-emulable graphs
We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph)-a problem raised already in [email protected]? thesis (1985) andExpand
And Fellows' Conjecture Yo'av Rieck and Yasushi Yamashita
In 1988 Fellows conjectured that if a finite, connected grap h dmits a finite planar emulator, then it admits a finite planar cover. We cons truct a finite planar emulator for K4,5−4K2. D. ArchdeaconExpand
Planar emulators conjecture is nearly true for cubic graphs
We prove that a cubic nonprojective graph cannot have a finite planar emulator unless it belongs to one of two very special cases (in which the answer is open). This shows that Fellows' planarExpand
Minor-minimal non-projective planar graphs with an internal 3-separation
TLDR
It is proved that a graph is projective planar if and only if it has no minor isomorphic to a graph from a list of 35 specific graphs. Expand
Algebraic and combinatorial aspects of incidence groups and linear system non-local games arising from graphs
To every linear binary-constraint system (LinBCS) non-local game, there is an associated algebraic object called the solution group. Cleve, Liu, and Slofstra showed that a LinBCS game has a perfectExpand
Decidability of regular language genus computation
TLDR
It is shown that a new family of regular languages on a two-letter alphabet having arbitrary high genus and the genus size can grow at least exponentially in size |L|, which implies that the planarity of a regular language is decidable. Expand
Mike Fellows: Weaving the Web of Mathematics and Adventure
  • J. A. Telle
  • Computer Science, Mathematics
  • The Multivariate Algorithmic Revolution and Beyond
  • 2012
This informal tribute in honor of Mike Fellows' 60th birthday is based on some personal recollections.
A Guide to the Discharging Method
We provide a “how-to” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify theExpand
Finite planar emulators for K4, 5-4K2 and K1, 2, 2, 2 and Fellows' Conjecture
TLDR
A finite planar emulator is constructed for K"4","5-4K"2", which provides a counterexample to Fellows' Conjecture and proves that Negami's Planar Cover Conjectured is true if and only if K"1","2","2,''2 admits no finitePlanar cover. Expand

References

SHOWING 1-10 OF 38 REFERENCES
A note on possible extensions of Negami's conjecture
TLDR
This paper suggests another formulation of Negami's conjecture that has a straightforward generalization to higher nonorientable surfaces, and provides some support for the generalized version of this conjecture. Expand
A note on possible extensions of Negami's conjecture
A graph H is a cover of a graph G, if there exists a mapping φ from V(H) onto V(G) such that for every vertex ν of G, φ maps the neighbors of ν in H bijectively onto the neighbors of φ(ν) in G.Expand
On possible counterexamples to Negami's planar cover conjecture
TLDR
It is shown that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture, and a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in the list. Expand
On possible counterexamples to Negami's planar cover conjecture
A simple graph H is a cover of a graph G if there exists a mapping Φ from H onto G such that Φ maps the neighbors of every vertex u in H bijectively to the neighbors of Φ (ν) in G. Negami conjecturedExpand
K4, 4 - e has no finite planar cover
TLDR
It is proved the non-existence of a finite planar cover of K4,4−e, which is a graph that has an embedding in the projective plane and no minor-minimal non-projective graph has a finitePlanar cover. Expand
Another two graphs with no planar covers
A graph H is a cover of a graph G if there exists a mapping Φ from V(H) onto V(G) such that Φ maps the neighbors of every vertex ν in H bijectively to the neighbors of Φ(ν) in G. Negami conjecturedExpand
A Kuratowski theorem for the projective plane
TLDR
Let I ( S ) denote the set of graphs, each with no valency 2 vertices, which are irreducible for S, and using this notation the authors state Kuratowski's theorem. Expand
On the parity of planar covers
TLDR
It is proved that if G is a planar graph that covers a nonplanar H, then the fold number must be even. Expand
Composite planar coverings of graphs
  • S. Negami
  • Computer Science, Mathematics
  • Discret. Math.
  • 2003
We shall prove that a connected graph G is projective-planar if and only if it has a 2n-fold planar connected covering obtained as a composition of an n-fold covering and a double covering for someExpand
The spherical genus and virtually planar graphs
  • S. Negami
  • Mathematics, Computer Science
  • Discret. Math.
  • 1988
TLDR
It is shown that G is virtually planar if and only if G is either planar or projective-planar and that sph( G ) = 1, 2 or ∞. Expand
...
1
2
3
4
...