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Generalized Hamiltonian Cycles Jakub Teska School of ITMS University of Ballarat, VIC 3353, Australia Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 1/27 Hamiltonian cycle Hamiltonian cycle is a cycle in a graph which

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 1/27

Generalized Hamiltonian Cycles

Jakub Teska

School of ITMS University of Ballarat, VIC 3353, Australia

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 2/27

Hamiltonian cycle

■ Hamiltonian cycle is a cycle in a graph which visits every vertex of the graph. ■ Decide whether a graph is hamiltonian is well known NP-Complete problem.

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Hamiltonian cycle

■ Hamiltonian cycle is a cycle in a graph which visits every vertex of the graph. ■ Decide whether a graph is hamiltonian is well known NP-Complete problem. ■ If a graph G is hamiltonian then G is 2-connected.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 3/27

Toughness

■ The toughness of a non-complete graph is t(G) = min( |S| c(G−S)), where the

minimum is to be taken over all nonempty vertex sets S, for which c(G − S) ≥ 2.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 4/27

Toughness

■ If a graph G is t-tough then G is ⌈2t⌉-connected.

Opposite implication is not true. There exist graphs with arbitrary large connectivity and arbitrary small toughness. Km,n for m ≥ n is n-connected but toughness t(Km,n) = n

m

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Necessary conditions

■ If a graph G is Hamiltonian then G is 1-tough

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 5/27

Necessary conditions

■ If a graph G is Hamiltonian then G is 1-tough ■ If toughness t(G) < 1 then G has no Hamiltonian cycle

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Sufficient conditions

Chv´ atal’s Conjecture : There exists a finite constant t0 such that every t0-tough

graph is hamiltonian. For many years the focus was on determining whether all 2-tough graphs are

  • hamiltonian. But in 2000 Bauer, Broersma and Veldman proved the following

theorem.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 6/27

Sufficient conditions

Chv´ atal’s Conjecture : There exists a finite constant t0 such that every t0-tough

graph is hamiltonian. For many years the focus was on determining whether all 2-tough graphs are

  • hamiltonian. But in 2000 Bauer, Broersma and Veldman proved the following

theorem.

■ For every ǫ > 0, there exists a ( 9 4 − ǫ)-tough graph without a Hamiltonian

cycle. To prove similar theorem to the Chvátal’s Conjecture we have to restrict our focus on some special classes of graphs.

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Chordal graphs

■ Graph is chordal if every cycle of length greater then three has a chord.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 7/27

Chordal graphs

■ Graph is chordal if every cycle of length greater then three has a chord. ■ Vertex x is simplicial vertex in G if NG(x)G is complete graph. ■ Assume that graph G is chordal. Then G has a simplicial vertex v and G − v

is chordal graph. Every chordal graph can be constructed from K3 just by recursive adding of new simplicial vertices.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 8/27

Chordal graphs

■ Every 18-tough chordal graph is Hamiltonian. (Chen et. al. 1997) ■ For every ǫ > 0, there exists a ( 7 4 − ǫ)-tough chordal graph without a

Hamiltonian cycle.(Bauer, Broersma and Veldman, 2000)

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Chordal graphs

■ Every 18-tough chordal graph is Hamiltonian. (Chen et. al. 1997) ■ For every ǫ > 0, there exists a ( 7 4 − ǫ)-tough chordal graph without a

Hamiltonian cycle.(Bauer, Broersma and Veldman, 2000)

■ Every chordal planar graph with t(G) > 1 is hamiltonian. (B˝

  • hme et. al.

1999)

■ There exists a sequence G1, G2, ... of 1-tough chordal planar graphs with c(Gi) |V (Gi)| → 0 as i → ∞.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 9/27

Sketch of the proof

■ If t(G) > 1 then G is 3-connected. Then degree of every vertex is at least

three.

■ If G is chordal planar graph, then G does not contain K5 as a subgraph and

therefor degree of every simplicial vertex is at most three. G can be constructed from K3 just by recursive adding of new simplicial vertices, but we can do it as follows: In every step we add set S of all simplicial vertices into the neighborhood of a simplicial vertex.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 10/27

Sketch of the proof

■ |S| < 3

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 10/27

Sketch of the proof

■ |S| < 3

Suppose that from graph Gi we get graph Gi+1 by adding set S of all simplicial vertices into the neibourhood of a simplicial vertex.

■ If Gi is hamiltonian then Gi+1 is hamiltonian.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 10/27

Sketch of the proof

■ |S| < 3

Suppose that from graph Gi we get graph Gi+1 by adding set S of all simplicial vertices into the neibourhood of a simplicial vertex.

■ If Gi is hamiltonian then Gi+1 is hamiltonian.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 10/27

Sketch of the proof

■ |S| < 3

Suppose that from graph Gi we get graph Gi+1 by adding set S of all simplicial vertices into the neibourhood of a simplicial vertex.

■ If Gi is hamiltonian then Gi+1 is hamiltonian.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 10/27

Sketch of the proof

■ |S| < 3

Suppose that from graph Gi we get graph Gi+1 by adding set S of all simplicial vertices into the neibourhood of a simplicial vertex.

■ If Gi is hamiltonian then Gi+1 is hamiltonian.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 11/27

k-walks

■ A k-walk in a graph G is a spanning closed walk which visits every vertex of

G at most k-times. This generalizes the notion of a Hamiltonian cycle because 1-walk in G is exactly a Hamiltonian cycle in G.

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k-walks

■ Every graph containing a k-walk is 1 k-tough.

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k-walks

■ Every graph containing a k-walk is 1 k-tough.

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k-walks

■ Every graph containing a k-walk is 1 k-tough.

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k-walks

■ Every graph containing a k-walk is 1 k-tough.

If t(G) < 1

k then G does not contain a k-walk.

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2-walks

■ Every 4-tough graph has a 2-walk. (Ellingham, Zha 2000)

This is similar theorem to the Chv´ atal’s Conjecture for 2-walks

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2-walks

■ Every 4-tough graph has a 2-walk. (Ellingham, Zha 2000)

This is similar theorem to the Chv´ atal’s Conjecture for 2-walks

■ For every ǫ > 0 and every k ≥ 1, there exists a ( 8k+1 4k(2k−1) − ǫ)-tough graph

with no k-walk. For k = 2 we get that there exists ( 17

24 − ǫ)-tough graph with no 2-walk.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 14/27

Idea of the proof

■ Every 4-tough graph has a 2-walk. (Ellingham, Zha 2000) ■ If G is 2-tough then G has a 2-factor.

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Idea of the proof

■ Every 4-tough graph has a 2-walk. (Ellingham, Zha 2000) ■ If G is 2-tough then G has a 2-factor.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 14/27

Idea of the proof

■ Every 4-tough graph has a 2-walk. (Ellingham, Zha 2000) ■ If G is 2-tough then G has a 2-factor.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 14/27

Idea of the proof

■ Every 4-tough graph has a 2-walk. (Ellingham, Zha 2000) ■ If G is 2-tough then G has a 2-factor.

Then Eulerian cycle in this graph coresponds to a 2-walk in the original graph.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 15/27

New result

■ Theorem : Every chordal planar graph with t(G) > 3 4 has a 2-walk.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 15/27

New result

■ Theorem : Every chordal planar graph with t(G) > 3 4 has a 2-walk.

Every simplicial vertex has degree 2 or 3.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 15/27

New result

■ Theorem : Every chordal planar graph with t(G) > 3 4 has a 2-walk.

Every simplicial vertex has degree 2 or 3. G can be constructed from K3 just by recursive adding of new simplicial vertices. From graph Gi we get graph Gi+1 by adding set S of all simplicial vertices into the neibourhood of a simplicial vertex.

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New result

■ Theorem : Every chordal planar graph with t(G) > 3 4 has a 2-walk.

Every simplicial vertex has degree 2 or 3. G can be constructed from K3 just by recursive adding of new simplicial vertices. From graph Gi we get graph Gi+1 by adding set S of all simplicial vertices into the neibourhood of a simplicial vertex. Now we have the following cases: I) Degree of x in Gi is two II) Degree of x in Gi is three A) T visits two edges incident with x in Gi B) T visits one edge incident with x in Gi

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Case I

■ If degree of x in Gi is two then |S| ≤ 2.

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Case I

■ If degree of x in Gi is two then |S| ≤ 2.

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 16/27

Case I

■ If degree of x in Gi is two then |S| ≤ 2.

Case I A

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Case I

■ If degree of x in Gi is two then |S| ≤ 2.

Case I A

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 16/27

Case I

■ If degree of x in Gi is two then |S| ≤ 2.

Case I A Case IB

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Case I

■ If degree of x in Gi is two then |S| ≤ 2.

Case I A Case IB

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 16/27

Case I

■ If degree of x in Gi is two then |S| ≤ 2.

Case I A Case IB

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Case II

■ If degree of x in Gi is three then |S| ≤ 4.

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Case II

■ If degree of x in Gi is three then |S| ≤ 4.

Case II A; |S| = 3

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Case II

■ If degree of x in Gi is three then |S| ≤ 4.

Case II A; |S| = 3

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 17/27

Case II

■ If degree of x in Gi is three then |S| ≤ 4.

Case II A; |S| = 3

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 17/27

Case II

■ If degree of x in Gi is three then |S| ≤ 4.

Case II A; |S| = 3

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 17/27

Case II

■ If degree of x in Gi is three then |S| ≤ 4.

Case II A; |S| = 3 Case IB; |S| = 3

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 17/27

Case II

■ If degree of x in Gi is three then |S| ≤ 4.

Case II A; |S| = 3 Case IB; |S| = 3

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 18/27

Case II

Case II A; |S| = 4

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Case II

Case II A; |S| = 4

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 18/27

Case II

Case II A; |S| = 4 Bad cases

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 19/27

Lower bound

Theorem : There exists an infinite class of 2-connected chordal planar graphs with toughness t(G) = 1

2 without a 2-walk.

x y v

  • u

u u u

1 2 3 4

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 20/27

Open problems

Conjectures:

■ There exists a finite constant t0 such that every t0-tough graph is hamiltonian. ■ Every 2-tough chordal graph is hamiltonian. ■ Every 1 k−1-tough graph has a k-walk. ■ Every 2-tough graph has a 2-walk. ■ Every 1-tough chordal graph has a 2-walk. ■ Every more then 1 2-tough chordal planar graph has a 2-walk.

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Trestles

■ For any integer r > 1, an r-trestle is a 2-connected graph F with maximum

degree ∆(F) ≤ r.

■ We say that a graph G has an r-trestle if G contains a spanning subgraph

which is an r-trestle.

■ A graph G is called K1,r-free if G has no K1,r as an induced subgraph.

1,3

K

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Trestles

Ryjᡠcek and Tkᡠc (2004) proved that

■ every 2-connected K1,3-free graph has a 3-trestle

They also conjectured that

■ every 2-connected K1,r-free graph has an r-trestle for every r ≥ 4.

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Trestles

Ryjᡠcek and Tkᡠc (2004) proved that

■ every 2-connected K1,3-free graph has a 3-trestle

They also conjectured that

■ every 2-connected K1,r-free graph has an r-trestle for every r ≥ 4.

  • Theorem. Every 2-connected K1,r-free graph has an r-trestle for every r ≥ 2.
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Proof - the main trick = good choice

  • 1. |V (T)| is maximal,

T

Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.

T

Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

T

Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

T

Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r

T

Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r

T

1

S Y A X X1

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Where is the vertex Y ?

T

1

S Y A X X1

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A

T

1

S Y A X X1

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A dT (X1) = r

T

1

S Y A X X1

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A dT (X1) = r

T

1

S Y X A

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A S1 S2 . . . Si.

T

S S S

1 2 3 n

S Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A S1 S2 . . . Si.

T’

S S S

1 2 3 n

S Y X

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Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A S1 S2 . . . Si. T ′ remains 2-connected

T’

S S S

1 2 3 n

S Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A S1 S2 . . . Si. T ′ remains 2-connected dT ′(X) = r − 1

T’

S S S

1 2 3 n

S Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 23/27

Proof - the main trick = good choice

  • 1. |V (T)| is maximal,
  • 2. |E(T)| is minimal.
  • 3. dT (X) + dT (Y ) is

minimal

  • 4. dT (X) ≥ dT (Y ).

dT (X) = r Y / ∈ S1 or Y = A S1 S2 . . . Si. T ′ remains 2-connected dT ′(X) = r − 1 - contradic- tion

T’

S S S

1 2 3 n

S Y X

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Jakub Teska, October 2, 2006 Generalized Hamiltonian Cycles - p. 24/27

Sharpness of the result

1 2 r−2 1 2 r−2 1 2 r−2

■ The example shows a K1,r-free graph having an r-trestle but no

(r − 1)-trestle for r ≥ 3.

■ The result of Theorem cannot be improved.

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Sharpness of the result

1 2 r−2 1 2 r−2 1 2 r−2

■ The example shows a K1,r-free graph having an r-trestle but no

(r − 1)-trestle for r ≥ 3.

■ The result of Theorem cannot be improved.

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Sharpness of the result

1 2 r−2 1 2 r−2 1 2 r−2

■ The example shows a K1,r-free graph having an r-trestle but no

(r − 1)-trestle for r ≥ 3.

■ The result of Theorem cannot be improved.

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More results on trestles...

■ A minimum-degree condition for the existence of an r-trestle was recently

proved by Jendrol’, Ryjᡠcek and Schiermeyer.

■ There is a polynomial algorithm for finding r-trestle in a given K1,r-free graph.

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More results on trestles...

■ A minimum-degree condition for the existence of an r-trestle was recently

proved by Jendrol’, Ryjᡠcek and Schiermeyer.

■ There is a polynomial algorithm for finding r-trestle in a given K1,r-free graph. ■ Every 2-edge-connected graph with maximum degree ∆ has a ⌈ ∆+1 2 ⌉-walk

(Kaiser, Kuˇ zel, Li, Wang; 2006)

■ Every r-trestle has an ⌈ r+1 2 ⌉-walk for any integer r ≥ 2. (R. K., J. T. ’05)

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The End

Thank You