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Invariant universality for projective planes [article]

Filippo Calderoni, Gianluca Paolini
2020 arXiv   pre-print
We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes  ...  We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly  ...  Definition 1), and by ⊑ pl and ∼ =pl the relation of embeddability and isomorphism on X pl , respectively. (3) We denote by X ppl the standard Borel space of countable non-Desarguesian projective planes  ... 
arXiv:1801.10107v3 fatcat:qj3jqgncw5fyphqnx2e5gsrvfu

Uniquely and faithfully embeddable projective-planar triangulations

Seiya Negami
1984 Journal of combinatorial theory. Series B (Print)  
Every 5connected projective-planar triangulation, not isomorphic to K, , is uniquely and faithful@ embeddable in a projective plane.  ...  In this paper, we 189 SEIYA NEGAMI shall give another sufficient condition for a projective-planar graph to be uniquely and faithfully embeddable in a projective plane. THEOREM 1.  ... 
doi:10.1016/0095-8956(84)90024-8 fatcat:rd7jsroejvgdbjqwit6c2nlvse

Page 6813 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews  
He shows that this conjecture for the projective plane is equivalent to Negami’s conjecture and proves that K7, Kg minus a perfect matching, and the Heawood graph Hj4, which is isomorphic to the dual of  ...  In 1986, Negami conjectured that if a graph G has a finite planar cover, then G is embeddable in the projective plane.  ... 

Page 2850 of Mathematical Reviews Vol. , Issue 83h [page]

1983 Mathematical Reviews  
a unital which is embeddable, and isomorphic with its dual, but not the set of absolute points of a polarity (in fact examples exist in each of the four known projective planes of order 9), (iii) a unital  ...  Piper [Graph theory and combinatorics (Milton Keynes, 1978), pp. 98-105, Pitman, San Francisco, Calif., 1979; MR 81i:05045] by exhibiting (i) a unital that is not embeddable in a projective plane, (ii)  ... 

Topological Graph Theory from Japan

Seiya NEGAMI
2001 Interdisciplinary Information Sciences  
This is a survey of studies on topological graph theory developed by Japanese people in the recent two decades and presents a big bibliography including almost all papers written by Japanese topological  ...  graph theorists.  ...  It is faithfully embeddable in the projective plane unless it is isomorphic to K 6 .  ... 
doi:10.4036/iis.2001.99 fatcat:6qxw75knznei5gjuzlfk6inxre

Page 46 of Mathematical Reviews Vol. , Issue 87a [page]

1987 Mathematical Reviews  
, if G is not isomorphic to Kg, then G is faithfully embeddable in a projective plane.”  ...  It is shown that if a projective-planar graph G is 5-connected and contains a subdivision of the complete graph Kg as a subgraph, then G is uniquely embeddable in a projective plane, and that, moreover  ... 

Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic [article]

Kengo Enami
2019 arXiv   pre-print
We shall characterize structures of a 3-connected 3-regular planar graph G embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings  ...  Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface.  ...  Acknowledgements The auther thanks anonymous referees for careful reading and their helpful comments which improved the quality of this paper.  ... 
arXiv:1806.11333v4 fatcat:prpn6af3uzdovbvmlgomhmwicm

Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

Kengo Enami
2018 Discrete Mathematics & Theoretical Computer Science  
We shall characterize structures of a 3-connected 3-regular planar graph G embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings  ...  Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface.  ...  Acknowledgements The auther thanks anonymous referees for careful reading and their helpful comments which improved the quality of this paper.  ... 
doi:10.23638/dmtcs-21-4-14 fatcat:vcdnu4tx6va2xi57ic5f4xk3ti

Enumeration of projective-planar embeddings of graphs

Seiya Negami
1986 Discrete Mathematics  
It will be shown that the number of equivalence classes of embeddings of a 3-connected nonplanar graph into a projective plane coincides with the number of isomorphism classes of planar double coverings  ...  of the graph and a combinatorial method to determine the number will be developed.  ...  For example, Ka.3 has a unique planar double covering C6 x/(2, as is mentioned in the last section, and hence it is uniquely embeddable in a projective plane.  ... 
doi:10.1016/0012-365x(86)90217-7 fatcat:vhb2ffxdizhztaq6rwrwgzhvce

Re-embedding of projective-planar graphs

Seiya Negami
1988 Journal of combinatorial theory. Series B (Print)  
In this paper, we characterize those projective-plane 3-connected graphs which admit re-embeddings in a projective plane different from their original one.  ...  The results will be applied for analysis of the uniqueness and faithfulness of the embedding of Sconnected projective-planar graphs.  ...  A Sconnected projective-planar graph which triangulates a projective plane is uniquely and faithfully embeddable in a projective plane unless it is isomorphic to K,.  ... 
doi:10.1016/0095-8956(88)90037-8 fatcat:vdviiqx3ebhl5eaxzgyoef3gw4

A Kochen-Specker system has at least 22 vectors (extended abstract)

Sander Uijlen, Bas Westerbaan
2014 Electronic Proceedings in Theoretical Computer Science  
Continuing their effort, we prove a restriction on the class of graphs we need to consider and develop a more practical decision procedure for embeddability to improve the lower bound to 22.  ...  The bottleneck in their search proved to be the sheer number of graphs on more than 17 vertices and deciding embeddability.  ...  McKay for discussing the feasibility of certain graph restrictions and to Judith van Stegeren 13 for drawing figures for the draft version of this paper.  ... 
doi:10.4204/eptcs.172.11 fatcat:ihyptvrzorcgdidaq3mektbyjm

A Kochen-Specker System Has at Least 22 Vectors

Sander Uijlen, Bas Westerbaan
2016 New generation computing  
Continuing their effort, we prove a restriction on the class of graphs we need to consider and develop a more practical decision procedure for embeddability to improve the lower bound to 22.  ...  The bottleneck in their search proved to be the sheer number of graphs on more than 17 vertices and deciding embeddability.  ...  McKay for discussing the feasibility of certain graph restrictions and to Judith van Stegeren 13 for drawing figures for the draft version of this paper.  ... 
doi:10.1007/s00354-016-0202-5 fatcat:vpypln3egrbbfi4xgnkbfkq6rq

Minimal decompositions of complete graphs into subgraphs with embeddability properties

L. W. Beineke
1969 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.  ...  Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other  ...  Place the 4r -1 vertices Vj consecutively on one semicircle a of the projective plane model aa. See Figure 5 for the case r = 3.  ... 
doi:10.4153/cjm-1969-109-4 fatcat:pdfbzu7scbbr5kuxch6p7idin4

Page 6744 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
With the help of this construc- tion he shows that there exist exactly two dimension pairs for which the following result holds: Every projectively embedded Grassmann space of this dimension pair is isomorphic  ...  The paper summarizes the author’s results describing (the obviously defined) adjacency graphs of chambers and walls for the configu- rations indicated in the title.  ... 

Page 12 of Mathematical Reviews Vol. , Issue 89M [page]

1989 Mathematical Reviews  
In this paper the author presents two new proofs for the problem of one isomorphism class.  ...  While Kuratowski’s theorem implies that M;(plane) contains exactly two graphs, M) (projective plane) consists of more than 100 graphs and M;(torus) more than 800 graphs.  ... 
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