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Hamiltonian Cycles on Symmetrical Graphs

Bridges 2004, Winfield KS. Hamiltonian Cycles on Symmetrical Graphs. Carlo H. Séquin EECS Computer Science Division University of California, Berkeley. Map of Königsberg. Can you find a path that crosses all seven bridges exactly once – and then returns to the start ?.

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Hamiltonian Cycles on Symmetrical Graphs

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  1. Bridges 2004, Winfield KS Hamiltonian Cycleson Symmetrical Graphs Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. Map of Königsberg • Can you find a path that crosses all seven bridges exactly once – and then returns to the start ? Leonhard Euler (1707-83) says: NO ! (1735)– because there are vertices with odd valence.

  3. START END • Hamiltonian Path:Visits all vertices once. • Hamiltonian Cycle:A closed Ham. Path. Definitions • Eulerian Path:Uses all edges of a graph. • Eulerian Cycle:A closed Eulerian Paththat returns to the start.

  4. This is a Test … (closed book!) • What Eulerian / Hamiltonian Path / Cycle(s)does the following graph contain ?

  5. Answer: • It admits an Eulerian Cycle !– but no Hamiltonian Path.

  6. Another Example … (extra credit!) • What paths/cycles exist on this graph? • No Eulerian Cycles: Not all valences are even. • No Eulerian Paths: >2 odd-valence vertices. • Hamiltonian Cycles? – YES! • = Projection of a cube (edge frame); Do other Platonic solids have Hamiltonian cycles ?

  7. The Platonic Solids in 3D • Hamiltonian Cycles ? • Eulerian Cycles ?

  8. The Octahedron • All vertices have valence 4. • They admit 2 paths passing through. • Pink edges form Hamiltonian cycle. • Yellow edges form Hamiltonian cycle. • The two paths are congruent ! • All edges are covered. • Together they form a Eulerian cycle. • Are there other (semi-)regular polyhedra for which we can do that ?

  9. Hamiltonian cycleon polyhedron edges. The Cuboctahedron • Flattened net ofcuboctahedronto show symmetry. Thecyanand theredcycles arecongruent (mirrored)!

  10. Larger Challenges • All these graphs have been planar … boring ! • Our examples had only two Hamiltonian cycles. • Can we find graphs that are covered by three or more Hamiltonian cycles ?  Graphs need to have vertices of valence ≥ 6. • Can we still make those cycles congruent ?  Graphs need to have all vertices equivalent. • Let’s look at complete graphs, i.e., N fully connected vertices.

  11. Complete Graphs K5, K7, and K9 • 5, 7, 9 vertices – all connected to each other. • Let’s only consider graphs with all even vertices,i.e., only K2i+1. K5 K7 K9 Can we make the i Hamiltonian cycles in each graph congruent ?

  12. The common Hamiltonian cycle for all K2i+1 Complete Graphs K2i+1 • K2i+1 will need i Hamiltonian cycles for coverage. • Arrange nodes with i-fold symmetry: 2i-gon  C2i • Last node is placed in center.

  13. Make Constructions in 3D … • We would like to have highly symmetrical graphs. • All vertices should be of the same even valence. • All vertices should be connected equivalently. • Graph should allow for some symmetrical layout in 3D space. • Where can we obtain such graphs? From 4D!(But don’t be afraid of the 4D source … This is really just a way of getting interesting 3D wire frames on which we can play Euler’s coloring game.

  14. The 6 Regular Polytopes in 4D From BRIDGES’2002 Talk

  15. Which 4D-to-3D Projection ?? • There are many possible ways to project the edge frame of the 4D polytopes to 3D.Example: Tesseract (Hypercube, 8-Cell) Cell-first Face-first Edge-first Vertex-first Use Cell-first: High symmetry; no coinciding vertices/edges

  16. C2 4D Simplex: 2 Hamiltonian Paths Two identical paths, complementing each other

  17. 4D Cross Polytope: 3 Paths • All vertices have valence 6 ! C3

  18. There are many different options: Hypercube: 2 Hamitonian Cycles

  19. C4 (C2) The Most Satisfying Solution ? • Each Path has its own C2-symmetry. • 90°-rotationaround z-axischanges coloron all edges.

  20. Natural to consider one C4-axis for replicating the Hamiltonian cycles. C4 The 24-Cell Is More Challenging • Valence 8: -> 4 Hamiltonian pathsare needed !

  21. This is a mess …hard to deal with ! Exploiting the concentric shells: 5 families of edges. 24-Cell: Shell-based Approach

  22. OUTER SHELL MIDDLE SHELL INNER SHELL This is the visitation schedule for one Ham. cycle. Shell-Schedule for the 24-Cell • The 24 vertices lie on 3 shells. • Pre-color the shells individuallyobeying the desired symmetry. • Rotate shells against each other.

  23. One Cycle – Showing Broken Symmetry Almost C2

  24. That is what I had to inspect for … C2-Symmetry  More than One Cycle! C2

  25. Path Composition

  26. 24-Cell: 4 Hamiltonian Cycles Aligned:  4-fold symmetry

  27. Why Shells Make Task Easier • Decompose problem into smaller ones: • Find a suitable shell schedule; • Prepare components on shells compatible with schedule; • Find a coloring that fits the schedule and glues components together,by “rotating” the shells and connector edges within the chosen symmetry group. • Fewer combinations to deal with. • Easier to maintain desired symmetry.

  28. C3* C3* C3* C3* Tetrahedral Symmetry on “0cta”-Shell • C3*-rotations that keep one color in place, cyclically exchange the three other ones:

  29. “Tetrahedral” Symmetry for the 24-Cell • All shells must have the same symmetry orientation- this reduces the size of the search tree greatly. • Of course such a solution may not exist … Note that the same-colored edges are disconnected !

  30. OUTER SHELL MIDDLE SHELL INNER SHELL Another Shell-Schedule for the 24-Cell • Stay on each shell for only one edge at a time: This is the schedule needed for overall tetrahedral symmetry !

  31. OUTER SHELL I/O-MIRROR MIDDLE SHELL INNER SHELL • We can also use inside / outside symmetry,so only 1/6 of the cycle needs to be found ! Exploiting C3-Symmetry for the 24-Cell • Only 1/3 of the cycle needs to be found.(C3-axis does not go through any edges, vertices) REPEATED UNIT

  32. INSIDE-OUTSIDE SYMMETRY One of the C3-symmetric Cycles

  33. CENTRAL CUBOCTA OUTER OCTA Pipe-Cleaner Models for the 24-Cell That is actually how I first found the tetrahedral solution ! INNER OCTA

  34. Rapid Prototyping Model of the 24-Cell • Noticethe 3-foldpermutationof colorsMade on the Z-corp machine.

  35. 3D Color Printer(Z Corporation)

  36. The Uncolored 120-Cell • 120-Cell: • 600 valence 4-vertices, 1200 edges • --> May yield 2 Hamiltonian cycles length 600.

  37. Brute-force approach for the 120-Cell Thanks to Mike Pao for his programming efforts ! • Assign opposite edges different colors (i.e., build both cycles simultaneously). • Do path-search with backtracking. • Came to a length of 550/600, but then painted ourselves in a corner !(i.e., could not connect back to the start). • Perhaps we can exploit symmetryto make search tree less deep.

  38. A More Promising Approach • Clearly we need to employ the shell-based approach for these monsters! • But what symmetries can we expect ? • These objects belong to the icosahedral symmetry group which has 6 C5-axes and 10 C3-axes. • Can we expect the individual paths to have 3-fold or 5-fold symmetry ?This would dramatically reduce the depth of the search tree !

  39. Simpler Model with Dodecahedral Shells Just two shells (magenta) and (yellow) Each Ham.-path needs 15 edges on each shell,and 10 connectors (cyan) between the shells.

  40. Dodecahedral Double Shell Colored by two congruent Hamiltonian cycles

  41. Physical Model of Penta-Double-Shell

  42. The 600-Cell • 120 vertices,valence 12; • 720 edges;  Make 6 cycles, length 120.

  43. Search on the 600-Cell • Search by“loop expansion”: • Replace an edge in the current pathwith the two other edges of a triangleattached to the chosen edge. •  Always keeps path a closed cycle ! • This quickly worked for finding a full cycle. • Also worked for finding 3 congruent cycles of length 120. • When we tried to do 4 cycles simultaneously, we got to 54/60 using inside/outside symmetry.

  44. Shells in the 600-Cell INNERMOST TETRAHEDRON Number of segments of each type in each Hamiltonian cycle INSIDE / OUTSIDE SYMMETRY OUTERMOST TETRAHEDRON CONNECTORS SPANNING THE CENTRAL SHELL

  45. Shells in the 600-Cell Summary of features: • 15 shells of vertices • 49 different types of edges: • 4 intra shells with 6 (tetrahedral) edges, • 4 intra shells with 12 edges, • 28 connector shells with 12 edges, • 13 connector shells with 24 edges. • Inside/outside symmetry • What other symmetries are there … ?

  46. Start With a Simpler Model … Specifications: • All vertices of valence 12 • Overall symmetry compatible with “tetra-6” • Inner-, outer-most shells = tetrahedra • No edge intersections • As few shells as possible … …. This is tricky … 

  47. Icosi-Tetrahedral Double-Shell (ITDS) Just 4 nested shells (192 edges): • Tetrahedron: 4V, 6E • Icosahedron: 12V, 30E • Icosahedron: 12V, 30E • Tetrahedron: 4V, 6E total: 32V CONNECTORS

  48. ITDS: The 2 Icosahedral Shells

  49. The Complete ITDS: 4 shells, 192 edges TETRA ICOSA ICOSA TETRA SHELLS CONNECTORS

  50. One Cycle on the ITDS SHELLS CONNECTORS TETRA ICOSA ICOSA TETRA

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