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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

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).