Shortness coefficient of cyclically 4-edge-connected cubic graphs - - PowerPoint PPT Presentation

▶

Oct 22, 2022 187 likes •671 views

Introduction Cyclically 4-edge-connected Future work Shortness coefficient of cyclically 4-edge-connected cubic graphs On-Hei S. Lo Jens M. Schmidt Nico Van Cleemput Carol T. Zamfirescu Combinatorial Algorithms and Algorithmic Graph Theory

SLIDE 1

Introduction Cyclically 4-edge-connected Future work

Shortness coefficient

cyclically 4-edge-connected cubic graphs

On-Hei S. Lo Jens M. Schmidt Nico Van Cleemput Carol T. Zamfirescu

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

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 1

SLIDE 2

Introduction Cyclically 4-edge-connected Future work

1 Introduction Definitions Known results

2 Cyclically 4-edge-connected cubic graphs The planar case Higher genera Bounded face length General cubic graphs

3 Future work

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 2

SLIDE 3

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Circumference

The circumference circ(G) is the length of a longest cycle.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 3

SLIDE 4

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Hamiltonicity

A graph G is hamiltonian if circ(G) = |V(G)|.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 4

SLIDE 5

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Hamiltonicity of classes of graphs

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)

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 5

SLIDE 6

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Hamiltonicity of classes of graphs

The first to construct a counterexample was Tutte in 1946

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 6

SLIDE 7

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Hamiltonicity of classes of graphs

Theorem (Tutte, 1956) Every 4-connected polyhedron is hamiltonian.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 7

SLIDE 8

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Hamiltonicity of classes of graphs

How far is a class of graphs from being hamiltonian?

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 8

SLIDE 9

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Shortness coefficient

The shortness coefficient of G is defined as ρ (G) = lim inf

G∈G

circ(G) |V(G)| with lim inf taken over all sequences of graphs Gn in G such that |V(Gn)| → ∞ for n → ∞.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 9

SLIDE 10

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Shortness coefficient

ρ (G) = lim inf

G∈G

circ(G) |V(G)| 0 ≤ ρ (G) ≤ 1 every graph in G is hamiltonian ⇒ ρ (G) = 1

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 10

SLIDE 11

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Known results

Theorem (Moon and Moser, 1963) The shortness coefficient of the class of 3-connected planar graphs is 0. Theorem (Tutte, 1956) The shortness coefficient of the class of 4-connected planar graphs is 1.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 11

SLIDE 12

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Known results

Theorem (Bondy and Simonovits, 1980) The shortness coefficient of the class of 3-connected cubic graphs is 0. Theorem (Walther, 1969) The shortness coefficient of the class of 3-connected cubic planar graphs is 0.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 12

SLIDE 13

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Cyclically k-edge-connected

A graph G is cyclically k-edge-connected if for every edge-cut S of G with less than k edges at most one component of G − S contains a cycle.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 13

SLIDE 14

Introduction Cyclically 4-edge-connected Future work Definitions Known results

Cyclically k-edge-connected

For k ∈ {1, 2, 3} being cyclically k-edge-connected and being k-connected are equivalent for cubic graphs. Ck is the class of cyclically k-edge-connected cubic graphs. CkP is the class of cyclically k-edge-connected planar cubic graphs.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 14

SLIDE 15

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Outline

1 Introduction Definitions Known results

2 Cyclically 4-edge-connected cubic graphs The planar case Higher genera Bounded face length General cubic graphs

3 Future work

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 15

SLIDE 16

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Known bounds

circ (G) ≥ 3

4|V(G)|

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 16

SLIDE 17

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Known bounds

Theorem (Grünbaum and Malkevitch, 1976) ρ(C4P) ≤ 76

Theorem (Lo and Schmidt, 2018) ρ(C4P) ≤ 52

Question ρ(C4P) ≤ 41

42?

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 17

SLIDE 18

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

Theorem (Lo, Schmidt, VC, and Zamfirescu) ρ(C4P) ≤ 37

Approach Find cyclically 4-edge-connected fragments such that (almost) any intersection with a cycle misses some vertices. Combine these fragments to construct an infinite family of graphs

btaining the bound in the limit.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 18

SLIDE 19

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Fragments and cycles

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 19

SLIDE 20

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

a d c b

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 20

SLIDE 21

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

a d c b

H − a is non-hamiltonian

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 21

SLIDE 22

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

a d c b

H − a is non-hamiltonian H − d is non-hamiltonian H − a − b is non-hamiltonian H − c − d is non-hamiltonian H − ab − cd is non-hamiltonian

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 22

SLIDE 23

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 23

SLIDE 24

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

F F F F F F F F F F F F

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 24

SLIDE 25

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

misses at least k − 2 vertices

F F F F F F F F F F F F

k copies of fragment misses at least k vertices

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 25

SLIDE 26

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

ρ (C4P) = lim inf

G∈C4P

circ(G) |V(G)| ≤ lim

k→∞

38k − (k − 2) 38k = 37 38

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 26

SLIDE 27

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Higher genus

Theorem (Lo, Schmidt, VC, and Zamfirescu) For every g ≥ 0, the shortness coefficient of the class of cyclically 4-edge-connected cubic graphs of genus g is at most 37

38.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 27

SLIDE 28

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Increasing the genus

F F F F F F F F F F F G

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 28

SLIDE 29

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A fragment with arbitrary genus

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 29

SLIDE 30

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A fragment with arbitrary genus

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 30

SLIDE 31

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A fragment with arbitrary genus

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 31

SLIDE 32

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A fragment with arbitrary genus

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 32

SLIDE 33

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A fragment with arbitrary genus

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 33

SLIDE 34

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Bounded face length

Theorem (Lo, Schmidt, VC, and Zamfirescu) For all ℓ ≥ 23, the shortness coefficient of the class of cyclically 4-edge-connected cubic plane graphs with faces of length at most ℓ is at most 45

46.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 34

SLIDE 35

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A second fragment

a b c d

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 35

SLIDE 36

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A second fragment

a b c d

H is not hamiltonian H − a is not hamiltonian H − d is not hamiltonian

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 36

SLIDE 37

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A second fragment

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 37

SLIDE 38

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

A new bound

Replacing each vertex of a 4-connected 4-regular planar graph on k vertices by this fragment results in a cyclically 4-edge-connected cubic planar graph in which each cycle spanning multiple fragments misses at least one vertex in each fragment. ρ (C4P) = lim inf

G∈C4P

circ(G) |V(G)| ≤ lim

k→∞

45k 46k = 45 46

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 38

SLIDE 39

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Bounded face length

5 vertices 3 vertices 5 vertices 10 vertices

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 39

SLIDE 40

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Bounded face length

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 40

SLIDE 41

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Bounded face length

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 41

SLIDE 42

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Bounded face length

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 42

SLIDE 43

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Bounded face length

23 23 23 23 23 23 23 23 20 18 20 11 11 18 20 20

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 43

SLIDE 44

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Increasing the genus

Theorem (Lo, Schmidt, VC, and Zamfirescu) For every g ≥ 0 and for every ℓ ≥ 23, the shortness coefficient of the class of cyclically 4-edge-connected cubic graphs of genus g with faces of length at most ℓ is at most 45

46.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 44

SLIDE 45

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

Increasing the genus

G 23 23 23 23 23 23 23 23 18 11 18 11 11 18 20 20

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 45

SLIDE 46

Introduction Cyclically 4-edge-connected Future work Planar Higher genera Bounded face length General

General cubic graphs

Theorem (Lo, Schmidt, VC, and Zamfirescu) Let G be a cyclically 4-edge-connected cubic graph on n vertices. Then ρ(C4) ≤ circ(G) − 2 n − 2 , and if there exist adjacent vertices v, w in G such that G − v − w is planar, then ρ(C4P) ≤ circ(G) − 2 n − 2 . Corollary ρ(C4) ≤ 7

8 and ρ(C4P) ≤ 39 40.

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 46

SLIDE 47

Introduction Cyclically 4-edge-connected Future work

Future work

3 4 ≤ ρ(C4P) ≤ 37 38

shrink the gap fragments are smallest possible missing more vertices

quartic graphs? quintic graphs?

On-Hei S. Lo, Jens M. Schmidt, Nico Van Cleemput, Carol T. Zamfirescu Shortness coefficient of cyclically 4-edge-connected cubic graphs 47