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The Number of Loopless 4-Regular Maps on the Projective Plane

Han Ren, Yanpei Liu
2002 Journal of combinatorial theory. Series B (Print)  
In this paper rooted loopless (near) 4-regular maps on surfaces such as the sphere and the projective plane are counted and exact formulae with up to three or four parameters for such maps are given.  ...  Several classical results on regular maps and one-faced maps are deduced.  ...  One of the anonymous referees suggested we add some examples such as those determined by the maps in Fig. 2 together with a table as one of the initial steps of the numerical verification of Theorems  ... 
doi:10.1006/jctb.2001.2064 fatcat:ibkyuahvnnaknfn43gxcffw5dq

4-Regular Maps on the Klein Bottle

Han Ren, Yanpei Liu
2001 Journal of combinatorial theory. Series B (Print)  
Also the number of rooted near-4-regular maps with exactly k nonroot-vertex loops on this surface is provided.  ...  In this paper the number of rooted near-4-regular maps on the Klein bottle is investigated and formulae with up to four parameters are given.  ...  examples such as those determined by the maps in Fig. 2 as one of the initial steps of the numerical verification of Theorem 2.  ... 
doi:10.1006/jctb.2000.2030 fatcat:hurpgfx4xjczhpif5ispcajoaa

Page 7046 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
Bhave (Pune) 2002j:05083 05C30 05A15 0SC10 57MI5 Ren, Han (PRC-ECNU; Shanghai); Liu, Yanpei (PRC-NJT; Beijing) The number of loopless 4-regular maps on the projective plane. (English summary) J.  ...  From the summary: “In this paper rooted loopless (near) 4-regular maps on surfaces such as the sphere and the projective plane are counted, and exact formulae with up to three or four parameters for such  ... 

Enumeration of 2 -connected Loopless 4 -regular Maps on the Plane

Han Ren, Yanpei Liu, Zhaoxiang Li
2002 European journal of combinatorics (Print)  
In this paper rooted (near-) 4-regular maps on the plane are counted with respect to the root-valency, the number of edges, the number of inner faces, the number of non-root vertex loops, the number of  ...  Several known results on 4-regular maps and trees of Tutte are also concluded. Finally, asymptotic formulae for the numbers of those types of maps are given.  ...  ACKNOWLEDGEMENTS The computations of the approximate resultants were carried with the aid of a symbolic algebra system (MAPLE) on a PENTIUM/II. The authors would like to thank Professor Gao Zhicheng  ... 
doi:10.1006/eujc.2001.0533 fatcat:qp7bfk7o7babhbt3xhtrkokc7m

4-edge-connected 4-regular maps on the projective plane

Shude Long, Han Ren
2020 Ars Mathematica Contemporanea  
Several known results on the number of 4-regular maps on the projective plane are also derived.  ...  Finally, using Darboux's method, a nice asymptotic formula for the numbers of this type of maps is given which implies that almost every (loopless) 4-regular map on the projective plane has a separating  ...  (9) rooted 2-connected 4-regular maps on the plane and the projective plane [30, 40] ; (10) rooted 4-edge-connected 4-regular maps on the plane [31] .  ... 
doi:10.26493/1855-3974.1615.979 fatcat:3vhj2mowurfkxltafp4k6m2lgq

Page 8806 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
(PRC-NJT; Beijing) Enumeration of 2-connected loopless 4-regular maps on the plane.  ...  In this paper the authors count rooted (near) 4-regular maps on the plane with respect to the root valency, the number of edges, the number of inner faces, the number of non-root vertex loops, the number  ... 

European Journal of Combinatorics Contents, Volume 26

2005 European journal of combinatorics (Print)  
. . . . . . . 271 Numbers 3-4 Tower of Hanoi graphs and Stern's diatomic sequence . . . . . . . . . . . . . . 693 COOLSAET, K., A distance regular graph with intersection array (21, 16, 8; 1, 4,  ...  WALSH, T.R., Counting unrooted loopless planar maps . . . . . . . . . . . . . . . . . . 651 KEY, J.D., MCDONOUGH, T.P. and MAVRON, V.C., Partial permutation decoding for codes from finite planes . . .  ... 
doi:10.1016/s0195-6698(05)00128-9 fatcat:fb5uilehgvgzxgmj7sfw4lm5hq

Contents of Volume 23

2002 European journal of combinatorics (Print)  
Loopless 4-regular Maps on the Plane . . . . . . . . . . . 93 SZEGEDY, C., On the Number of 3-Edge Colorings of Cubic Graphs . . 113 TANAKA, H., A Four-Class Subscheme of the Association  ...  619 SCARABOTTI, F., On a Lemma of Gromov and the Entropy of a Graph . 631 Number 6 CAUCHIE, S., DE CLERCK, F. and HAMILTON, N., Full Embeddings of (α, β)-Geometries in Projective Spaces . .  ... 
doi:10.1006/eujc.2002.0040 fatcat:6zapgfe62naipplawwaptralda

Page 3586 of Mathematical Reviews Vol. , Issue 92g [page]

1992 Mathematical Reviews  
B 53 (1991), no. 1, 130-142. in this paper the rooted loopless triangular maps on the projec- tive plane are enumerated by number of vertices.  ...  Palmer (1-MIS) 92g:05101 05C30 05A15 05C10 57M99 Gao, Zhi-Cheng (1-UCSD) The number of rooted 2-connected triangular maps on the projective plane. J. Combin. Theory Ser.  ... 

Index to Volume 23

2002 European journal of combinatorics (Print)  
., Enumeration of 2-connected Loopless 4-regular Maps on the Plane . SAMET-VAILLANT, A. Y., Algebras of Linear Growth, the Kurosh-Levitzky Problem and Large Independent Sets .  ...  ., On the Number of 3-Edge Colorings of Cubic Graphs . . 113 TANAKA, H., A Four-Class Subscheme of the Association Scheme Coming from the Action of P G L(2, 4 f ) TANIGUCHI, H., d-Dimensional Dual Hyperovals  ... 
doi:10.1006/eujc.2002.0041 fatcat:77afpkkn35dbrnwpnndkb64tvi

Positively Hyperbolic Varieties, Tropicalization, and Positroids [article]

Felipe Rincón, Cynthia Vinzant, Josephine Yu
2021 arXiv   pre-print
Along the way, we also give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.  ...  We prove that positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type A hyperplane arrangement defined by x_i = x_j, in which the maximal cones satisfy a non-crossing  ...  Then the numbers s 1 s 2 , s 1 s 3 , s 1 s 3 , s 2 s 3 , s 2 s 4 , s 2 s 4 all lie on the same line through 0.  ... 
arXiv:1907.08545v3 fatcat:ft3bcvvdmbhc3ouswxybdddoru

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

2000 Mathematical Reviews  
We show that a triangulation G of the sphere, the projective plane, the torus or the Klein bottle is 1-loosely tight if and only if both the independence number and the diameter of G do not exceed 2.  ...  Using this result, we classify all l-loosely tight triangulations of the projective plane.”  ... 

Page 1446 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews  
.; White, Arthur T. (1-WMI; Kalamazoo, MI) Regular imbeddings of finite projective planes.  ...  Summary: “An imbedding of the bipartite incidence graph of PG(2,7) into a closed orientable 2-manifold is called regular if its automorphism group induces a regular action on the set of flags.  ... 

Coloring-flow duality of embedded graphs

Matt DeVos, Luis Goddyn, Bojan Mohar, Dirk Vertigan, Xuding Zhu
2004 Transactions of the American Mathematical Society  
In 1996, Youngs discovered that every quadrangulation of the projective plane has chromatic number 2 or 4, but never 3.  ...  A map φ : E(G) → R is a tension if for every circuit C ⊆ G, the sum of φ on the forward edges of C is equal to the sum of φ on the backward edges of C.  ...  There is an embedding K of K 6 with edge-width 3 on the projective plane. The dual K * is Petersen's graph.  ... 
doi:10.1090/s0002-9947-04-03544-5 fatcat:5c6ykx4synf65iarm4ipd6zbuy

Eulerian digraphs and toric Calabi-Yau varieties [article]

Paul de Medeiros
2010 arXiv   pre-print
We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples.  ...  We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.  ...  I am also grateful to the members of the Institute for the Physics and Mathematics of the Universe (IPMU) for their kind hospitality during the conception of this work.  ... 
arXiv:1011.2963v1 fatcat:k4njlbyhbrd7tgee3x25rnysfm
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