Non-Hamiltonian and Non-Traceable Regular 3-Connected Planar Graphs - - PowerPoint PPT Presentation

▶
non hamiltonian and non traceable regular 3 connected
SMART_READER_LITE
LIVE PREVIEW

Non-Hamiltonian and Non-Traceable Regular 3-Connected Planar Graphs - - PowerPoint PPT Presentation

Apr 22, 2023 257 likes •677 views

Introduction Quartic Quintic Conclusion Non-Hamiltonian and Non-Traceable Regular 3-Connected Planar Graphs Nico Van Cleemput Carol T. Zamfirescu Combinatorial Algorithms and Algorithmic Graph Theory Department of Applied Mathematics,

slide-1
SLIDE 1

Introduction Quartic Quintic Conclusion

Non-Hamiltonian and Non-Traceable Regular 3-Connected Planar Graphs

Nico Van Cleemput Carol T. Zamfirescu

Combinatorial Algorithms and Algorithmic Graph Theory Department of Applied Mathematics, Computer Science and Statistics Ghent University

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 1

slide-2
SLIDE 2

Introduction Quartic Quintic Conclusion

1

Introduction Definitions Cubic Quartic Quintic Summary

2

Quartic Upper bound c4 Lower bound c4 Upper bound p4

3

Quintic Upper bound p5

4

Conclusion Summary Future work

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 2

slide-3
SLIDE 3

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Here, a polyhedron is a planar 3-connected graph. The word “regular” is used exclusively in the graph-theoretical sense of having all vertices of the same degree. By Euler’s formula, there are k-regular polyhedra for exactly three values of k: 3, 4, or 5.

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 3

slide-4
SLIDE 4

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Let ck be the order of the smallest non-hamiltonian k-regular polyhedron. Let pk be the order of the smallest non-traceable k-regular polyhedron.

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 4

slide-5
SLIDE 5

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – hamiltonicity

Tait conjectured in 1884 that every cubic polyhedron is hamiltonian. The conjecture became famous because it implied the Four Colour Theorem (at that time still the Four Colour Problem)

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 5

slide-6
SLIDE 6

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – hamiltonicity

The first to construct a counterexample (of order 46) was Tutte in 1946

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 6

slide-7
SLIDE 7

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – hamiltonicity

Lederberg, Bosák, and Barnette (pairwise independently) described a smaller counterexample having 38 vertices.

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 7

slide-8
SLIDE 8

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – hamiltonicity

After a long series of papers by various authors (e.g., Butler, Barnette, Wegner, Okamura), Holton and McKay showed that all cubic polyhedra on up to 36 vertices are hamiltonian. Theorem (Holton and McKay, 1988) c3 = 38

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 8

slide-9
SLIDE 9

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – traceability

Balinski asked whether cubic non-traceable polyhedra exist Brown and independently Grünbaum and Motzkin proved the existence of such graphs Klee asked for determining p3

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 9

slide-10
SLIDE 10

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – traceability

In 1970 T. Zamfirescu constructed this cubic non-traceable planar graph on 88 vertices

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 10

slide-11
SLIDE 11

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Cubic polyhedra – traceability

Based on work of Okamura, Knorr improved a result of Hoffmann by showing that all cubic planar graphs on up to 52 vertices are traceable. Theorem (Knorr, 2010 and Zamfirescu, 1970) 54 ≤ p3 ≤ 88

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 11

slide-12
SLIDE 12

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Quartic polyhedra – hamiltonicity

Following work of Sachs from 1967 and Walther from 1969, Zaks proved in 1976 that there exists a quartic non-hamiltonian polyhedron of order 209. The actual number given in Zaks’ paper is false, as pointed out in work of Owens — therein the correct number can be found.

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 12

slide-13
SLIDE 13

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Quartic polyhedra – hamiltonicity

Theorem (Sachs, 1967) If there exists a non-hamiltonian (non-traceable) cubic polyhedron of

  • rder n, then there exists a non-traceable (non-hamiltonian) quartic

polyhedron on 9n

2 vertices.

On page 132 of Bosák’s book it is claimed that converting the Lederberg-Bosák-Barnette graph with this method gives a quartic non-hamiltonian polyhedron of order 161. However, the correct number should be 38 × 9

2 = 171.

Theorem (Sachs, 1967 combined with Bosák, 1990) c4 ≤ 171

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 13

slide-14
SLIDE 14

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Quartic polyhedra – traceability

Zaks showed that p4 ≤ 484 Using Sachs’ theorem on Zamfirescu’s 88-vertex graph gives a non-traceable quartic polyhedron on 396 vertices. Theorem (Sachs, 1967 combined with Zamfirescu, 1970) p4 ≤ 396

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 14

slide-15
SLIDE 15

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Quintic polyhedra

Previous work includes papers by Walther, as well as Harant, Owens, Tkᡠc, and Walther. Zaks showed that c5 ≤ 532 and p5 ≤ 1232. Owens proved that c5 ≤ 76 and p5 ≤ 128. Theorem (Owens, 1980) c5 ≤ 76 p5 ≤ 128

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 15

slide-16
SLIDE 16

Introduction Quartic Quintic Conclusion Definitions Cubic Quartic Quintic Summary

Summary

Hamiltonicity Traceability Cubic c3= 38 54 ≤p3≤ 88 Quartic c4≤ 171 p4≤ 396 Quintic c5≤ 76 p5≤ 128

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 16

slide-17
SLIDE 17

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound hamiltonicity

Theorem (Van Cleemput and Zamfirescu, 2017+) c4 ≤ 39

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 17

slide-18
SLIDE 18

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound hamiltonicity

strong weak

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 18

slide-19
SLIDE 19

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound hamiltonicity

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 19

slide-20
SLIDE 20

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound hamiltonicity

F1 F2 F3 x y z

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 20

slide-21
SLIDE 21

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound hamiltonicity

F1 F2 F3 x y z

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 21

slide-22
SLIDE 22

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity

Check all quartic polyhedra for being hamiltonian.

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 22

slide-23
SLIDE 23

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – hamiltonicity check

Simple backtracking algorithm that tries to construct a cycle from the first vertex.

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 23

slide-24
SLIDE 24

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – hamiltonicity check

16 17 18 19 20 21 22 2−9 2−5 2−1 23 27 No optimisation Start reachable No unreachable vertices Faces on two sides Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 24

slide-25
SLIDE 25

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – hamiltonicity check

16 17 18 19 20 21 22 2−9 2−5 2−1 23 27 No optimisation Start reachable No unreachable vertices Faces on two sides No plane graph Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 24

slide-26
SLIDE 26

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

18 20 22 24 26 28 30 0% 50% 100% Generating graphs Checking hamiltonicity Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 25

slide-27
SLIDE 27

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 26

slide-28
SLIDE 28

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 27

slide-29
SLIDE 29

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 28

slide-30
SLIDE 30

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 29

slide-31
SLIDE 31

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 30

slide-32
SLIDE 32

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

10 15 20 25 30 0% 50% 100% Not constructed Type 1 Type 2 Remaining Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 31

slide-33
SLIDE 33

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity – generation

25 26 27 28 29 30 31 Without filtering With filtering Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 32

slide-34
SLIDE 34

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Lower bound hamiltonicity

Theorem (Van Cleemput and Zamfirescu, 2017+) c4 ≥ 34 Vertices Time 25 7.7 minutes 26 34.9 minutes 27 2.7 hours 28 13.2 hours 29 2.7 days 30 13.8 days 31 71.3 days 32 1.0 years 33 5.5 years

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 33

slide-35
SLIDE 35

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound traceability

Theorem (Van Cleemput and Zamfirescu, 2017+) p4 ≤ 78

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 34

slide-36
SLIDE 36

Introduction Quartic Quintic Conclusion Upper bound c4 Lower bound c4 Upper bound p4

Upper bound traceability

21 × 4 − 6 = 78 vertices

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 35

slide-37
SLIDE 37

Introduction Quartic Quintic Conclusion Upper bound p5

Upper bound traceability

Theorem (Van Cleemput and Zamfirescu, 2017+) p5 ≤ 120

β α β α α α α α

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 36

slide-38
SLIDE 38

Introduction Quartic Quintic Conclusion Upper bound p5

Upper bound traceability

α β

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 37

slide-39
SLIDE 39

Introduction Quartic Quintic Conclusion Summary Future work

Summary

Hamiltonicity Traceability Cubic c3 = 38 54 ≤p3 ≤ 88 Quartic c4 ≤ 171 p4≤ 396 34 ≤ c4 ≤ 39 34 ≤ p4≤ 78 Quintic c5 ≤ 76 p5≤ 128 38 ≤c5 ≤ 76 38 ≤p5 ≤ 120

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 38

slide-40
SLIDE 40

Introduction Quartic Quintic Conclusion Summary Future work

Future work

c4 ≥ 35 ? Hamiltonicity for quintic case Lower bounds for traceability

Nico Van Cleemput, Carol T. Zamfirescu Non-Hamiltonian and Non-Traceable Regular Polyhedra 39