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16. CLASSICAL CONFIGURATIONS

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Pappus Graph on Torus. 10, 17, 18, 13, 12, 11. 8, 15, 16, 17, 10, 9 ... It contains the Pappus configuration. It contains also the M bius-Kantor configuration. ... – PowerPoint PPT presentation

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Title: 16. CLASSICAL CONFIGURATIONS


1
16. CLASSICAL CONFIGURATIONS
2
Fano plane (73).
  • We can reconstruct (73) from the matrx M.
  • Columns are homogeneous coordinates in F2.
  • ltijkgt det (Mi Mj Mk)
  • ijk form a line if and only if ltijkgt 0.

3
Heawood Graph in Torus
  • On the left there is a hexagonal embedding of the
    Heawood graph in torus.
  • Its dual is a triangular embedding of K7 in S2.

4
Menger Graph on Torus
  • On the left there is a hexagonal embedding of the
    Heawood graph in torus.
  • Its dual is a triangular embedding of K7 in S2.

5
Menger Graph on Torus
  • Menger graph is K7 and has a triangular embedding
    in torus.

6
Tuttes 8-Cage Construction (III)
  • By gluing appropriately the leaves of the tree on
    the left to the midpoints of the edges of the
    cube on the right one obtains Tuttes 8-Cage.
  • Cubic graph
  • Bipartite graph
  • Girth 8
  • Diameter 4.

7
Question
  • Q. If we subdivide the edges of K_4 we may attach
    the tree on the left to it in such a way that we
    avoid quadrangles. What graph is produced in ths
    way?

8
Similar Question
  • Same for the S(K2,2,2) and the tree. First layer
    antipodal edges, second layer main squares of the
    octahedron. Truncate vertices of valence 4.

9
Möbius-Kantor Configuration Revisited
  • Möbius-Kantor configuration is the only (83)
    configuration. Its Levi graph is the generalized
    Petersen graph G(8,3).
  • The configuration has no geometric realization.

10
Complex Coordinatization of (94,123)
11
A Z3 coordinatization of (134) PG(2,3)
12
Exercises
  • N1. Show that by deleting any column of the
    matrix from the previous slide a coordinatization
    of (123,94) is obtained.
  • N2. Determine the homogeneous coordinates of the
    9 lines from the previous problem.
  • N3.Show that by deleting any column of the matrix
    for (94,123) a coordinatization of (83) is
    obtained.
  • N4. Given the Levi graph G(8,3) of (83),
    determine the Levi graph of (94,123).

13
Möbius-Kantor Configuration Revisited
14
Menger graph of Möbius-Kantor Configuration
  • Menger graph of this configuration is depleated
    K8 DK8 K8 4K2
  • Vertices represent configuration points while
    triangles represent lines.

15
Möbius-Kantor Graph in Double Torus
  • Möbius-Kantor graph in double torus .
  • The embedding is octagonal.
  • The map is regular.

16
Möbius-Kantor Graph in Double Torus
  • Möbius-Kantor graph in double torus gives rise to
    the embedding of the Menger graph DK8 in the same
    surface with 8 triangles and 6 quadrilaterals.
  • By adding 4 missing edges we get an embedding of
    K_8 in double torus with all triangles, except
    two quadrilaterals.

17
The Dual
  • The dual graph is S2(K4).
  • Let G be any graph. Recall that S(G) is the
    subdivision graph.
  • Sk(G) is obtained from S(G) by multiplying the
    original vertices of G k times.

18
Pappusova konfiguracija
  • Pappusova (93) konfiguracija sestoji iz devetih
    tock in devetih premic. Tockam lahko pripiemo
    homogene koordinate (a,b,c), pa tudi premicam
    lahko pripiemo homogene koordinate p,q,r pri
    cemer incidenco doloca zveza apbqcr0.
  • Ta primer lahko interpretiramo kot zgled
    ortogonalne reprezentacije grafov, pri katerih
    uv implicira r(u) r(v).

19
Pappus Graph on Torus
  • 10, 17, 18, 13, 12, 11
  • 8, 15, 16, 17, 10, 9
  • 7,12, 13, 14, 15, 8
  • 4, 11, 12, 7, 6, 5
  • 3,4, 5, 16, 15, 14
  • 2, 9, 10, 11, 4, 3
  • 1, 2, 3, 14, 13, 18
  • 1, 18, 17, 16, 5, 6
  • 1, 6, 7, 8, 9, 2

20
Three (93) Configurations
21
Three (93) Configurations
  • They are all combinatorially self-polar.
  • Pappus (red)
  • Cyclic (green)
  • Non-cyclic (yellow?).

22
Three (93) Configurations
  • 5, 11, 14, 7, 15, 16, 12, 6
  • 4, 10, 18, 17, 11, 5
  • 3, 9, 17, 18, 12, 16, 8, 13, 10, 4
  • 2, 8, 16, 15, 9, 3
  • 1, 2, 3, 4, 5, 6
  • 1, 6, 12, 18, 10, 13, 14, 11, 17, 9, 15, 7
  • 1, 7, 14, 13, 8, 2

23
Three (93) Configurations
  • 10, 16, 15, 11, 17, 18
  • 8, 18, 17, 9, 14, 13
  • 7, 15, 16, 12, 13, 14
  • 4, 5, 6, 12, 16, 10
  • 3, 9, 17, 11, 5, 4
  • 2, 3, 4, 10, 18, 8
  • 1, 2, 8, 13, 12, 6
  • 1, 6, 5, 11, 15, 7
  • 1, 7, 14, 9, 3, 2

24
Exercises
  • N1. Show that each (9_3) configuration is
    combinatorially self-polar.
  • N2. Determine the groups of automorphisms and
    extended automorphisms.
  • N3. Show that the genus of two configurations is
    1 while the genus of the third one is 2. Make
    models!
  • N4. Determine the three Menger graphs and their
    duals on the minimal surfaces.
  • N4. Prove that the complements of the three
    Menger graphs are respectively C9, C6 C3, 3C3.

25
Menger and Levi - Pappus
26
Menger and Levi Non-Cyclic
27
Menger and Levi - Cyclic
28
Again - Shaken
29
Menger and Its Complementof G(10,3)
30
Genus of G(10,3) is 2.
  • 6, 7, 17, 20, 13, 16
  • 5, 6, 16, 19, 12, 15
  • 4, 5, 15, 18, 11, 14
  • 3, 13, 20, 10, 9, 8, 7, 6, 5, 4
  • 2, 3, 4, 14, 17, 7, 8, 18, 15, 12
  • 1, 2, 12, 19, 9, 10
  • 1, 10, 20, 17, 14, 11
  • 1, 11, 18, 8, 9, 19, 16, 13, 3, 2

31
Affine plane of order 3
  • (94,123) configuration is the affine plane of
    order 3.
  • It contains the Pappus configuration.
  • It contains also the Möbius-Kantor configuration.

32
Clebschev estkotnik
33
Clebschev estkotnik e enkrat
34
Clebschev graf
35
Hiperkocka Q4
36
Clebschev graf e enkrat
37
Grayev graf G
  • Najmanji kubicni graf, ki je povezavno
    tranzitiven in ni vozlicno tranzitiven ima 54
    vozlic in je znan pod imenom Grayev graf.
    Oznacili ga bomo z G.
  • Ker ima oino 8, je Levijev graf dveh najmanjih
    dualnih, tockovno, premicno in praporno
    tranzitivnih, ne sebidualnih (273)-konfiguracij.

38
Grayeva konfiguracija
  • Ciklicno risanje obeh konfiguracij, ki izhajata
    iz Grayevega grafa.
  • Obe risbi prikazujeta probleme v zvezi z risanjem
    z ravnimi crtami. Na obeh opazimo lane
    incidence, ki jih v kombinatornih konfiguracijah
    ni. Ce se elimo izogniti lanim incidencam
    bodisi zgubimo 9-merno simetrijo bodisi izgubimo
    ravne crte.

39
Grayeva konfiguracija e enkrat
  • Slika na levi prikazuje mnogo boljo sliko
    Grayeve konfiguracije. Pomaga nam tudi lociti med
    Grayevo konfiguracijo in dualno Grayevo
    konfiguracijo.
  • Risba na levi pokae, da je Mengerjev graf M
    Grayeve konfiguracije izomorfen kartezicnemu
    produktu treh trikotnikov K3 ? K3 ? K3 .

40
Mengerjev in dualni Mengerjev graf
  • Lepa reprezentacija iz prejnje prosojnice olaja
    izbiro med Grayevo konfiguracijo in njenim dualom
    (ter med Mengerjevim grafom M K33 in dualnim
    Mengerjevim grafom D.)

41
Rod produkta K3 ? K3 ? K3
  • Pred leti so dokazali, da je g(K3 ? K3 ? K3 )
    7. Optimalno vloitev so konstruirali Mohar,
    Pisanski, koviera in White. Shematicno je
    prikazana na levi.

42
Optimalna vloitev
  • Optimalna vloitev s prejnje prosojnice ima
    nekaj zelo lepih lastnosti
  • Njen dual je dvodelni.
  • Vloitev uporabi vseh 27 trikotnikov grafa za
    lica
  • Ce pobarvamo lica z dvema barvama, je vseh 27
    trikotnikov pobarvanih z isto barvo.
  • Od tod pa sledi zelo pomembna ugotovitev Tocke
    Grayeve konfiguracije ustrezajo vozlicem,
    premice pa trikotnikom vloitve. Incidenca je
    seveda dolocena s povezavami.

43
Grayev graf lahko vloimo v sklenjeno
orientabilno ploskev roda 7
  • Ce obrimo originalna vozlica in vpeljemo nova
    vozlica v teicih trikotnikov in poveemo vsako
    teice z oglici, dobimo na ta nacin ravno
    Grayev graf.
  • Posledica Grayev graf je mogoce vloiti v isto
    ploskev!

44
Grayev graf lahko vloimo v sklenjeno
orientabilno ploskev roda 7
  • Ce obrimo originalna vozlica in vpeljemo nova
    vozlica v teicih trikotnikov in poveemo vsako
    teice z oglici, dobimo na ta nacin ravno
    Grayev graf.
  • Posledica Grayev graf je mogoce vloiti v isto
    ploskev!

45
Spodnja meja
  • Zgornja meja je potemtakem 7. Da je tudi spodnja
    meja 7, sledi iz naslednjega rezultata
  • Trditev Naj bo L Levijev in M Mengerjev graf
    neke (v3) konfiguracije C, tedaj velja ocena g(M)
    ? g(L).
  • Dokaz Zacnemo z optimalno vloitvijo grafa L. e
    opisani proces lahko obrnemo (in vpeljemo
    trikotnike). Tako dobimo vloitev grafa M v isto
    ploskev.

46
Dualni Mengerjev graf D
  • e na nekaj moramo opozoriti, namrec na dualni
    Mengerjev graf D, gi se ga da po isti logiki
    vloiti v sklenjeno ploskev roda 7. Ni teko
    videti, da je tudi ta graf s slike na levi
    zanimiv. Je namrec Cayleyev graf poldirektnega
    produkta ciklicnih grup Z3 ? Z9.
  • Lahko ga tudi opiemo kot Z9-krovni graf nad
    bazicnim grafom z naslednje prosojnice.

47
Napetostni graf
  • Dualni Mengerjev graf je Z9 krovni graf nad
    napetostnim grafom na levi. Napetosti so seveda
    iz grupe Z9.
  • Lahko ga tudi predstavimo kot Cayleyev graf z
    naslednje prosojnice.

2
-1
-4
4
1
2
1
4
-2
48
Holtov Graf
  • 4-valentni Holtov graph H je vpet podgraf grafa D
    na 27 vozlicih in je najmanji 1/2-locno
    tranzitivni graf. To pomeni, da je H najmanji
    vozlicno in povezavno, ne pa locno tranzitiven
    graf. Prikazuje ga slika na levi.

49
Holtov graf - ponovno
  • 4-valentni Holovt graf H je vpet podgraf grafa D.
    Iz D ga dobimo z odtranitvijo 2-faktorja 3C9.
    Natancneje H je Z9-krov nad zelenim grafov na
    levi z odstranjenimi tremi rdecimi zankami.

2
-1
-4
4
1
2
1
4
-2
50
Nekaj prezentacij grupe Z3 ? Z9
  • Grafa H in D sta Cayleyeva grafa grupe Z3 ? Z9.
  • Z3 ? Z9 lta, b a9 b3 1, b-1ab a2gt
  • D ltx, y, z x9 y9 z9 1, y-1xy x2,
    y-1zy z2, x-1yx y2, x-1zx z2, z-1xz
    x2, z-1yz y2gt
  • H dobimo iz D ce iz prezentacije odstranimo
    kateregakoli od x,y,z.

51
Nereeni problemi
  • Koliken je rod grafa D? Koliken je rod grafa H?
    Rod grupe Z3 ? Z9 je poznan g(Z3 ? Z9)
    4. Po drugi strani smo dokazali, da dopuca D
    vloitev v sklenjeno ploskev roda 7. Zato velja
  • 4 ? g(D) ? 7
  • 4 ? g(H) ? 7
  • V prvem primeru bo najbr laje izboljati
    spodnjo, v drugem pa zgornjo mejo.

52
Balabanova 10-kletka
  • Na levi vidimo Balabanovo 10-kletko. To je
    najmanji kubicni graf oine 10. Ima 70 vozlic,
    vidimo tudi ocitno simetrijo.
  • Kletka poseduje tudi Hamiltonov cikel. To vidimo
    npr. iz LCF kode zanjo
  • -9, -25, -19, 29, 13, 35, -13, -29, 19, 25, 9,
    -29, 29, 17, 33, 21, 9, -13, -31, -9, 25, 17, 9,
    -31, 27, -9, 17, -19, -29, 27, -17, -9, -29, 33,
    -25, 25, -21, 17, -17, 29, 35, -29, 17, -17, 21,
    -25, 25, -33, 29, 9, 17, -27, 29, 19, -17, 9,
    -27, 31, -9, -17, -25, 9, 31, 13, -9, -21, -33,
    -17, -29, 29

53
Preostali 10-kletki
  • Ob Balabanovi obstajata e dve 10-kletki. Druga
    je bolj simetricna od tretje.
  • (-29, -19, -13, 13, 21, -27, 27, 33, -13, 13,
    19, -21, -33, 29)5

54
Preostali 10-kletki
  • Ob Balabanovi obstajata e dve 10-kletki. Tretja
    je najman simetricna.
  • 9, 25, 31, -17, 17, 33, 9, -29, -15, -9, 9, 25,
    -25, 29, 17, -9, 9, -27, 35, -9, 9, -17, 21, 27,
    -29, -9, -25, 13, 19, -9, -33, -17, 19, -31, 27,
    11, -25, 29, -33, 13, -13, 21, -29, -21, 25, 9,
    -11, -19, 29, 9, -27, -19, -13, -35, -9, 9, 17,
    25, -9, 9, 27, -27, -21, 15, -9, 29, -29, 33, -9,
    -25.

55
10-kletke
  • Vse tri 10-kletke so hamiltonske, zato smo jih
    lahko opisali z LCF kodo.
  • Grupe avtomorfizmov imajo rede 80, 120, 24.
  • Literatura T.P., M. Boben, D. Maruic, A.
    Orbanic The 10-cages and derived Configurations,
    Discrete Math. 2003 (v tisku).
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