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Two New Classes of Hamiltonian Graphs

Two New Classes of Hamiltonian Graphs. Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki Joint work with Esther Arkin and Joseph Mitchell Applied Math and Statistics, Stony Brook University. Induced Graph. Subset S of R 2 vertices: S

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Two New Classes of Hamiltonian Graphs

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  1. Two New Classes of Hamiltonian Graphs Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki Joint work with Esther Arkin and Joseph Mitchell Applied Math and Statistics, Stony Brook University

  2. Induced Graph Subset S of R2 • vertices: S • edge (i,j) if |i – j| = 1

  3. Square Grid Graph • Subset S of Z2 • Solid grid • no “holes” • all bounded faces – unit squares

  4. Hamiltonicity of Square Grids • NP-complete in general [Itai, Papadimitriou, and Szwarcfiter ’82] • Solid grids • polynomial [Umans and Lenhart ’96]

  5. Tilings • Square grid • unit squares

  6. Tilings • Square grid • unit squares • Triangular grid • unit equilateral triangles

  7. Triangular Grid Graph Subset S vertices: S • edge (i,j) if |i – j| = 1 Hole: bounded face ≠ unit equilateral ∆

  8. Solid Triangular Grid No holes all bounded faces – unit equilateral triangles

  9. Previous Work • HamCycle Problem • NP-complete in general • Solid grids • always Hamiltonian • no deg-1 vertices The only non-Hamiltonian solid triangular grid

  10. Local Cut Single vertex whose removal decreases number of holes Solid ) No local cuts Our result: Triangular grids without local cuts are Hamiltonian

  11. Idea • B: • Cycle around the outer boundary • Cycles around holes’ boundaries • Use modifications • cycles go through all internal vertices • Exists “facing” rhombus • no local cuts = graph is “thick” • merge facing cycles • Decrease number of cycles • Get Hamiltonian Cycle

  12. L-modification

  13. V-modification

  14. Z-modification

  15. Priority: L , V , Z • L • V • Z

  16. Wedges • Sharp • 60o turn • Wide • 120o turn

  17. The Main Lemma Until B passes through ALL internal vertices • either L, V, or Z may be applied small print: unless G is the Star of David

  18. Internal vertex v not in B • A neighbor u is in B • Crossed edges • not in B • o.w. – apply L

  19. How is u visited? WLOG, 1 is in B

  20. s L cannot be applied s is in B How is s visited?

  21. Sharp Wedge Z s V s

  22. Wide Wedge L cannot be applied t is in B

  23. s Deja Vu Rhombus • edge of B • vertex not in B • vertex in B Unless • t is a wide wedge • modification! • welcome new vertex to B

  24. Another Wide Wedge Yet Another vertex • Yet Another rhombus Yet Another wide wedge

  25. And so on… Star of David!

  26. Cycle Cover → HamCycle • Cycles around the outer boundary • Cycles around holes’ boundaries • Use modifications • cycles go through all internal vertices • Exists “facing” rhombus • no local cuts = graph is “thick” • merge facing cycles • Decrease number of cycles • Get Hamiltonian Cycle

  27. Hamiltonian Cycles in High-Girth Graphs

  28. HamCycle Problem is NP-complete • Classic • Girth? • 4 [GJ] • 3 [CLRS] • NP-complete [Garey, Johnson, Tarjan’76] • planar • cubic • girth-5 Higher girth?

  29. Multi-Hamiltonicity • 1 HC 2 HCs cubic [Smith], any vert – odd-deg [Thomason’78] r-regular,r > 300 [Thomassen’98], r > 48 [Ghandehari and Hatami] 4-regular? conjecture [Sheehan’75] maxdeg ≥f( maxdeg/mindeg )[Horak and Stacho’00] bipartite, mindeg in a part = 3 [Thomassen’96] • 1 HC exp(maxdeg) HCs [Thomassen’96] • bipartite • 1 HC exp(girth) HCs [Thomassen’96] cubic or bipartite, mindeg in a part = 4 Planar maxdeg 3, high-girth? >1 HC? Small # of HCs?

  30. Our Contribution Planar maxdeg 3 arbitrarily large girth • HamCycle Problem is NP-complete • Exactly 3 HamCycles arbitrarly large # of vertices

  31. The Other Tiling: Infinite Hexagonal Grid • Induced graphs • hexagonal grids Is HamCycle Problem NP-hard for hexagonal grids?

  32. Attempt to Show NP-Hardness • Same idea as for square and triangular grids [Itai, Papadimitriou, and Szwarcfiter ‘82, Papadimitriou and Vazirani ’84, PAM’06] • HamCycle Problem • undirected planar bipartite graphs • max deg 3 G0 Embed 0o, 60o, 120o segments

  33. (Try to) Embed in Hex Grid

  34. Edges – Tentacles

  35. Traversing Tentacles

  36. Cross path connects adjacent nodes

  37. Return path returns to one of the nodes

  38. White Node Gadget

  39. Middle Vertex: 2 edges…

  40. Middle Vertex: 2 edges…

  41. Induces 2 cross, 1 return path

  42. Induces 2 cross, 1 return path

  43. Induces 2 cross, 1 return path

  44. Black Node Gadget

  45. Middle Vertex: 2 edges…

  46. Middle Vertex: 2 edges…

  47. Induces 2 cross, 1 return path

  48. Induces 2 cross, 1 return path

  49. Induces 2 cross, 1 return path

  50. Return Path Starts at white node Closes at black node

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