local properties of graphs and the hamilton cycle problem
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Local Properties of Graphs and the Hamilton Cycle Problem Johan de - PowerPoint PPT Presentation

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Local Properties of Graphs and the Hamilton Cycle Problem Johan de Wet 1 , 2 and Marietjie Frick 1 1 University of Pretoria 2 DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Bucharest 2018 Johan de Wet et al. (UP,

  1. The HCP for LT graphs with maximum degree 6 z 1 z 4 z i V(G’) z 2 z 5 Z i is the corresponding Graph G’ z 6 z 3 node in G Z 1 Z 4 Z 2 Graph G Z 5 Z 3 Z 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 11 / 31

  2. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  3. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  4. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  5. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  6. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  7. Locally hamiltonian (LH) graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  8. Locally hamiltonian (LH) graphs Smallest connected nonhamiltonian LH graph has order 11 and ∆ = 8 (Pareek et al. 1983). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  9. Locally hamiltonian (LH) graphs Smallest connected nonhamiltonian LH graph has order 11 and ∆ = 8 (Pareek et al. 1983). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  10. Locally hamiltonian (LH) graphs Smallest connected nonhamiltonian LH graph has order 11 and ∆ = 8 (Pareek et al. 1983). HCP NP-complete for ∆ = 9 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  11. Hamilton Cycle Problem for LH graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  12. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  13. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Chv´ atal’s proof is valid for ∆ ≥ 12. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  14. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Chv´ atal’s proof is valid for ∆ ≥ 12. Theorem (van Aardt et al. 2016) If G is a connected LH graph with ∆( G ) ≤ 6, then G is hamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  15. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Chv´ atal’s proof is valid for ∆ ≥ 12. Theorem (van Aardt et al. 2016) If G is a connected LH graph with ∆( G ) ≤ 6, then G is hamiltonian. There exist connected LH graphs with maximum degree 8 that are nonhamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  16. Locally Hamiltonian Graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  17. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  18. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  19. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then (a) G is LH . Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  20. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then (a) G is LH . (b) If G 1 and G 2 are planar, then so is G . Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  21. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then (a) G is LH . (b) If G 1 and G 2 are planar, then so is G . (c) If G is hamiltonian, so are both G 1 and G 2 . Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  22. LH Graphs - the Hamilton Cycle Problem Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  23. LH Graphs - the Hamilton Cycle Problem Theorem The HCP for LH graphs with ∆ ≥ 9 is NP-complete. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  24. LH Graphs - the Hamilton Cycle Problem Theorem The HCP for LH graphs with ∆ ≥ 9 is NP-complete. We will use the same approach as for LT graphs. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  25. LH Graphs - the Hamilton Cycle Problem Theorem The HCP for LH graphs with ∆ ≥ 9 is NP-complete. We will use the same approach as for LT graphs. v 1 H D v 2 x 2 v 3 u 3 x 3 x 1 u 2 u 1 (a) (b) The graph H is locally hamiltonian and nonhamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  26. LH Graphs - the Hamilton Cycle Problem Graph H is combined with two copies of graph D to create the graph F : Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 17 / 31

  27. LH Graphs - the Hamilton Cycle Problem Graph H is combined with two copies of graph D to create the graph F : F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 17 / 31

  28. LH Graphs - the Hamilton Cycle Problem Vertices and Nodes and edges in G’ borders in G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 18 / 31

  29. LH Graphs - the Hamilton Cycle Problem Vertices and Nodes and edges in G’ borders in G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 18 / 31

  30. LH Graphs - the Hamilton Cycle Problem Vertices and Nodes and edges in G’ borders in G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 18 / 31

  31. LH Graphs - the Hamilton Cycle Problem z 1 z 4 z i V(G’) z 5 Graph G’ z 2 Z i is the corresponding z 6 z 3 node in G Z 4 Z 1 Graph G Z 2 Z 5 Z 3 Z 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 19 / 31

  32. LH Graphs - the Hamilton Cycle Problem F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 20 / 31

  33. LH Graphs - the Hamilton Cycle Problem F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 20 / 31

  34. LH Graphs - the Hamilton Cycle Problem F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 20 / 31

  35. Locally 2-nested hamiltonian (L2H) graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  36. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  37. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Smallest connected nonhamiltonian (L2H) graph has order 13 and ∆ = 10. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  38. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Smallest connected nonhamiltonian (L2H) graph has order 13 and ∆ = 10. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  39. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Smallest connected nonhamiltonian (L2H) graph has order 13 and ∆ = 10. HCP NP-complete for ∆ = 13 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  40. L2H Graphs - the Hamilton Cycle Problem z 1 z 4 z i V( G’) Graph G’ z 2 z 5 Z i is the corresponding z 6 z 3 node in G Z 1 Z 4 Graph G Z 5 Z 2 Z 6 Z 3 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 22 / 31

  41. Locally Hamilton-connected graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  42. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  43. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Smallest connected nonhamiltonian (LHC) graph has order 15 and ∆ = 11. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  44. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Smallest connected nonhamiltonian (LHC) graph has order 15 and ∆ = 11. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  45. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Smallest connected nonhamiltonian (LHC) graph has order 15 and ∆ = 11. HCP NP-complete for ∆ = 15 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  46. LHC Graphs - the Hamilton Cycle Problem The graph G ’ The graph G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 24 / 31

  47. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  48. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  49. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). A graph is closed-locally Chv´ atal-Erd¨ os if α ( � N [ v ] � ) ≤ κ ( � N [ v ] � ) for any v ∈ V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  50. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). A graph is closed-locally Chv´ atal-Erd¨ os if α ( � N [ v ] � ) ≤ κ ( � N [ v ] � ) for any v ∈ V ( G ). A cl-LCE graph is 1-tough (Chen et al. 2013). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  51. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). A graph is closed-locally Chv´ atal-Erd¨ os if α ( � N [ v ] � ) ≤ κ ( � N [ v ] � ) for any v ∈ V ( G ). A cl-LCE graph is 1-tough (Chen et al. 2013). It is not known if cl-LCE graphs are hamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  52. Discussion Table: The values of key parameters for various local properties. LC LT LH L2H LHC cl-LCE LCE Minimum n ( G ) if G is not 1-tough 5 7 11 13 15 N/A N/A Minimum ∆( G ) if G is not 1-tough 4 5 8 10 11 N/A N/A HCP is NP-complete for ∆( G ) at least 5 6 9* 13* 15* ? ? Minimum degree of local connectedness 1 1 2 3 3 1 2 *It is not known whether these values are best possible. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 26 / 31

  53. Discussion Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  54. Discussion We can generalize the concept of L2H graphs to LkH graphs. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  55. Discussion We can generalize the concept of L2H graphs to LkH graphs. Locally ( k + 1)-connected Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  56. Discussion We can generalize the concept of L2H graphs to LkH graphs. Locally ( k + 1)-connected The HCP for LkH graphs is NP-complete for any k ≥ 1. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  57. Discussion We can generalize the concept of L2H graphs to LkH graphs. Locally ( k + 1)-connected The HCP for LkH graphs is NP-complete for any k ≥ 1. The important variable is the relationship between the local connectivity and local independence number. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  58. Discussion Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 28 / 31

  59. Discussion Oberly-Sumner Conjecture: A connected graph that is locally k -connected and K 1 , k +2 -free is hamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 28 / 31

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