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Hamiltonian Cycles in High-Girth Graphs

Hamiltonian Cycles in High-Girth Graphs. Hardness of HamCycle Problem. Classic Girth? 4 [GJ] 3 [CLRS] NP-complete [Garey, Johnson, Tarjan’76] planar cubic 3-connected girth-5 girth > 5 avg.deg < 3 “tight” Higher girth?.

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Hamiltonian Cycles in High-Girth Graphs

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  1. Hamiltonian Cycles in High-Girth Graphs

  2. Hardness of HamCycle Problem • Classic • Girth? • 4 [GJ] • 3 [CLRS] • NP-complete [Garey, Johnson, Tarjan’76] • planar • cubic • 3-connected • girth-5 • girth > 5 avg.deg < 3 • “tight” Higher girth?

  3. 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

  4. Small Degree, Large Girth • 1 HC exp(g) HCs [Thomassen’96] cubic or bipartite, mindeg in a part = 4 girth g Not planar girth > 5 avg deg < 3 Planar maxdeg 3, high-girth? >1 HC? Small # of HCs?

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

  6. The Reduction • Same idea as for square grids [Itai, Papadimitriou, and Szwarcfiter ‘82, Papadimitriou and Vazirani ‘84] • Hamiltonian Cycle • undirected planar bipartite graphs • max deg 3 G0 Embed 0o, 60o, 120o segments

  7. (Try to) Embed in Hex Grid

  8. Edges – Tentacles

  9. Traversing Tentacles

  10. Cross path connects adjacent nodes

  11. Return path returns to one of the nodes

  12. White Node Gadget

  13. Middle Vertex: 2 edges…

  14. Middle Vertex: 2 edges…

  15. Induces 2 cross, 1 return path

  16. Induces 2 cross, 1 return path

  17. Induces 2 cross, 1 return path

  18. Black Node Gadget

  19. Middle Vertex: 2 edges…

  20. Middle Vertex: 2 edges…

  21. Induces 2 cross, 1 return path

  22. Induces 2 cross, 1 return path

  23. Induces 2 cross, 1 return path

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

  25. HC in G HC in G0 Any node gadget adjacent to 2 cross paths 1 return path • Edges of G0 in HC Cross paths • Edges of G0 not in HC Return paths from white nodes

  26. Ham Cycle is NP-hard for Hex Grid? No… didn’t show how to turn a tentacle Can’t turn with these tentacles

  27. No Longer in a Hex Grid

  28. Subdivide (Shown) Edges Imagine: adjacent deg-2 vertices connected by length-g path

  29. Short Cycles? Not within a gadget any cycle uses a shown edge which is length-g path

  30. Short Cycles Through >1 Node Gadgets? Make tentacles loooong > g “wiggles”

  31. Girth g+2 Graph • Planar • turning tentacle • no longer an issue • not in a hex grid • Maxdeg 3 • Non-bipartite • white-node gadget already

  32. HC in G HC in G0 Any node gadget adjacent to 2 cross paths 1 return path • Edges of G0 in HC Cross paths • Edges of G0 not in HC Return paths from white nodes

  33. Theorem 1 For any g ≥ 6 HamCycle is NP-hard in planar deg ≤ 3 non-bipartite girth-g graphs

  34. Multi-Hamiltonicity • Planar • Bipartite • Maxdeg 3

  35. Exactly 3 HamCycles

  36. Theorem 2 For any g ≥ 6 exists planar deg ≤ 3 non-bipartite girth-g graph with exactly 3 HamCycles

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