Symmetries and Covers of Discrete Objects 2016 / Rydges Hotel, Queenstown, New Zealand / 14-19 February 2016
Faithful embedding of graphs on closed surfaces
Yokohama Na onal University, Japan Seiya Negami When and how the faithfulness was born Faithful embedding on surfaces
Aut(M(G)) Aut(G)
M(G) G
Aut(M (G)) = Aut(G) : a faithful embedding Hassler Whitney
Theorem 11.5 Every 3-connected planar graph is uniquely embeddable on the sphere. Hassler Whitney [W13] A set of topological invariants for graphs, Trans. Amer. Math. Soo. 34 (1933), 231-235. [W11] Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932), 150-168. Re-embedding of planar graphs
Rotation around two vertices Arrangement of blocks
Every 3-connected planar graph is uniquely embedded on the sphere. (H.Whitney, 1932) Uniqueness of embedding…
a c a c 1 2 3 1 2 3 not equivalent b b c a c a 3 1 not congruent 3 1 c a a c 1 2 3 a c 2 2 b not equivalent a c 3 1 congruent c a not faithful 2 Uniqueness of embedding…
f1 f1 G F2 G F2 ∃ ∃ ∃ h σ h f 2 F2 G F2 equivalent f2 congruent
f unique = unique + faithful G F2 ∀ ∃ σ h G 2 F f faithful Theorem 5.7 A graph G is 3-connected if and only if G is a wheel or can be obtained from a wheel by a sequence of edge-addition and 3-vertex-splitting
William T. Tutte [T13] A theory of 3-connected graphs, Indag. Math. 23 (1961), 441-455. Re-embeddings and minors No.1 ∃h’
Edge Vertex contraction splitting ∃h My doctoral thesis
• Uniqueness and faithfulness of embedding of graphs on closed surfaces
Tokyo Institute of Technology, 1985 Opening a new world… Thurston’s Lecture Note Gross and Tucker’s book
Jonathan L. Gross, Thomas W. Tucker Dover Publica ons, 1987 Coverings of a graph... 3 5 1
There exists a bijection between 1 5 6 4 2 the neighborhoods 2 6 of corresponding vertices. 3 4 4 3
4-fold, regular 6 2 2 6 4 1 having a group5 action
1 2 6 5 1 2 1 5 3 3 4 4 3 3 4 6 5 6 2 1 5 1 2 3-fold, irregular 3 4
5 6 Covering spaces and planarity
1 1 3
2 4 4 2 4 2 5 5 5
3 3 1
K5 a planar double covering of K5
• What graph has a planar covering? The projective plane and a double covering...
faithfully embeddable！ embed
the sphere a planar double covering
embed
a nonplanar graph the projective plane
• There is a bijection between planar double coverings and embeddings to the projective plane of a 3-connected nonplanar graph. Elliptic 2-Orbifolds
{id} Z Chapter 13 2
Polyhedral group＋
Zn Dn
Poluhedral group
Dn Proposal of “1-2-∞ Conjecture”
• A connected graph G can be embeddable on the projective plane if and only if G has a finite planar regular covering. 1-fold or 2-fold
• Can we omit “regular”?
⇒ 1-2-∞ conjecture
S. Negami, The spherical genus and virtually planar graphs, Discrete Math. 70 (1988), 159-168. Planar Cover Conjecture (Negami, 1986)
• A connected graph G can be embedded on the projective plane if and only if G has a finite planar covering.
• Necessity is clear, but sufficiency is still open.
S. Negami, The spherical genus and virtually planar graphs, Discrete Math. 70 (1988), 159-168. Last monster...
A2 = K1,2,2,2
• If this graph does not have any finite planar covering, then Planar Cover Conjecture is true. Harvest! Dis nguishing chroma c numbers…
• How many colors do we need to destroy the symmetry of a given graph completely?
χ D (G) ■ Dis nguishing chroma c number The minimum number of colors in dis nguishing colorings.
χ D (Q3 ) = 4 A dis nguishing 4-coloring Graphs on closed surfaces
• For any triangula on G on a closed surface, Negami Aut(G) χ D (G) ≤ O(g) 2012
• For any polyhedral map M(G) on a closed surface,
χ D (M (G)) ≤ χ(G) + 2 Aut(M(G)) unless G is isomorphic to Kn,n,n for n=2,3,4,5. • For any polyhedral bipar te map M(G) on a closed surface,
if |V(G)| > 20. χ D (M (G)) ≤ 3 Negami + Tucker 2013+ Distinguishing coloring on 3-regular maps
• Theorem Any 3-regular map on a closed surface is nearly distinguishing 3-colorable unless it is isomorphic to one of the following three:
– The cube Q3 on the sphere
– K3,3 on the torus with three hexagonal faces
– K4,4 − 4K2 on the torus with four hexagonal faces
Seiya Negami 3-Regular maps on closed surfaces are nearly distinguishing 3-colorable with few exceptions, Graph & Combin. (2015).
Encountered this… y d c
z w a b a
x y x c d faithfully embedded！ w z a Q3 M (K4,4 − 4K2 ) Faithful embedding of planar graphs
• Theorem Any 3-connected planar graph can be embedded faithfully on a suitable orientable closed surface other than the sphere with few exceptions. SIGMAP 2014 Faithful embedding of planar graphs
• Theorem Any 3-connected planar graph can be embedded faithfully on a suitable orientable closed surface other than the sphere unless it is one of the followings: • The tetrahedron • The octahedron • The dodecahedron • The truncated tetrahedron • The truncated dodecahedron • The truncated icosahedron • The icosidodecahedron The tetrahedron on the projective plane
3 4 1 2
1 2
4 3 4 3
(13), (24) (12)(3)(4)
Aut(M (K4 )) = Aut(K4 ) Petri dual
1 1
4 4
3 2 3 2
123 142 134 243 1243 2341 3142 Non-bipartitel graphs
• Any non-bipartite 3-connected planar graph can be embedded faithfully on a suitable nonorientable closed surface.
1 1 Möbius band
4 4
3 2 3 2 Faithful embeddings on nonorientable surfaces
• Theorem If a 3-connected planar graph G has a face such that an odd number of edges along its boundary cycle are equivalent, then G can be faithfully embedded on a suitable nonorientable closed surface. TGT26, 2014
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● Examples having faithful embeddings For further studies
• Classify all 3-connected planar graphs which have no faithful embedding on nonorientable closed surfaces. • Find a method to determine the genera of closed surfaces where a given 3-connected planar graph can be faithfully embedded. • Faithful genus… Thank you for your a en on!