Pancyclicity of 4-connected, Claw-free, P10-free Graphs - Morris.pdfPancyclicity How do we show this is true? Build Up 1 Show G has a 3-cycle. 2 Show that if G has an s-cycle, then

The Hamiltonian ProblemPancyclicity

Pancyclicity of 4-connected, Claw-free, P10-free

Graphs

Timothy Morris1

Michael FerraraPaul Wenger

University of Colorado Denver

May 14, 2011

1Partially supported by NSF Grant # 0742434

UCD GK12 Transforming Experiences Project

Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs


Definitions

A hamiltonian cycle is a cycle that contains every vertex

A graph G is hamiltonian if it has a hamiltonian cycle.



Forbidden Subgraphs

Definition (H-free)

A graph G is said to be H-free if it does not contain a copy of H

as an induced subgraph.

Claw

Paw



Forbidden Subgraphs

Definition (H-free)



Claw

Paw



Forbidden Subgraphs

Definition (H-free)



Claw

Paw



Forbidden Subgraphs

Theorem (Goodman, Hedetniemi, 1974)

If G is a 2-connected graph that is claw-free and paw-free, then G

is hamiltonian.



Proof of Goodman, Hedetniemi

Consider a largest cycle and a neighbor not on the cycle.

Claw

Paw





Claw

Paw





Claw

Paw





Claw

Paw





Claw

Paw



Forbidden Subgraphs

Theorem (Duffus, Gould, Jacobson, 1982)

Let G be a claw-free net-free graph.

1 If G is connected, then G contains a hamiltonian path.

2 If G is 2-connected, then G is hamiltonian.

Net



Forbidden Subgraphs

Conjecture (Matthews, Sumner, 1984)

Every 4-connected claw-free graph is hamiltonian.



Forbidden Subgraphs

Conjecture (Matthews, Sumner, 1984)

Every 4-connected claw-free graph is hamiltonian.

Theorem (Kaiser, Vrana, 2010)

Every 5-connected claw-free graph with minimum degree at least 6

is hamiltonian.



Another Conjecture on 4-connected graphs

Conjecture (Thomassen, 1986)

Every 4-connected line graph is hamiltonian.




Conjecture (Thomassen, 1986)

Every 4-connected line graph is hamiltonian.

Theorem (Ryjacek, 1997)

Thomassen’s Conjecture is equivalent to the Matthews-Sumner

Conjecture.




Conjecture (Plummer, 1993)

Every 4-connected 4-regular claw-free graph is hamiltonian.




Conjecture (Plummer, 1993)

Every 4-connected 4-regular claw-free graph is hamiltonian.

Theorem (Plummer, 1993)

Plummer’s Conjecture is equivalent to the Matthews-Sumner

Conjecture.



Forbidden Subgraphs of 2-connected graphs

Theorem (Faudree, Gould, 1997; extending Bedrossian 1991)

Let X and Y be connected graphs such that X ,Y 6= P3 and let G

be a 2-connected graph of order n ≥ 10.

G is {X ,Y }-free implies G is hamiltonian if and only if X is the

claw and Y is an induced subgraph of one of the following.

P6N(1, 1, 1) N(2, 1, 0) N(3, 0, 0)



Forbidden Subgraphs of graphs with higher connectivity

Theorem ( Luczak, Pfender, 2004)

Every 3-connected claw-free P11-free graph is hamiltonian.






Theorem (Lai, Xiong, Yan, Yan, 2009)

Every 3-connected claw-free N(8, 0, 0)-free graph is hamiltonian.






Theorem (Lai, Xiong, Yan, Yan, 2009)

Every 3-connected claw-free N(8, 0, 0)-free graph is hamiltonian.

Theorem (Pfender, 2005)

Every 4-connected claw-free -free graph is hamiltonian.



Pancyclic Graphs

Definition (pancyclic)

A graph G on n vertices is said to be pancyclic if it contains a

cycle of each length from 3 to n.



Pancyclic Graphs






Pancyclic Graphs






Pancyclic Graphs






Pancyclic Graphs






Pancyclic Graphs






Bondy’s Metaconjecture

Conjecture (Bondy, 1971)

Almost any non-trivial condition on a graph G that guarantees G

is hamiltonian also guarantees G is pancyclic possibly with a small

number of exceptional graphs.




Theorem (Faudree, Gould, 1997)


be a 2-connected graph of order n ≥ 10.

G is {X ,Y }-free implies G is pancyclic if and only if X is the claw

and Y is an induced subgraph of one of the following.

P6N(2, 0, 0)



A Matthews, Sumner Like Conjecture for Pancyclicity?

Theorem (Brandt, Favaron, Ryjacek, 2000)

For every k ≥ 2 there exists a k-connected claw-free graph that is

not pancyclic.




Theorem (Gould, Luczak, Pfender, 1997)


be a 3-connected graph.

G is {X ,Y }-free implies G is pancyclic if and only if X is the claw

and Y is an induced subgraph of one of the following.

P7 LN(2, 1, 1) N(2, 2, 0) N(3, 1, 0) N(4, 0, 0)



Problem (Gould, 2010)

Characterize the pairs of forbidden subgraphs that imply a

4-connected graph is pancyclic.



Path Results so far

Hamiltonicity

1 2-connected claw-freeP6-free(Bedrossian, 1991)

2 3-connected claw-freeP11-free( Luczak, Pfender, 2004)

Pancyclicity

1 2-connected claw-freeP6-free(Faudree, Gould, 1997)

2 3-connected claw-freeP7-free(Gould, Luczak, Pfender,1997)



Intuition



Intuition



The Goal

Thought (Ferrara, M, Wenger)

Every 4-connected, claw-free, P10-free graph is pancyclic.



How do we show this is true?

Build Up

1 Show G has a 3-cycle.

2 Show that if G has ans-cycle, then G has ans + 1-cycle.

Build Down

1 Show G has a hamiltoniancycle.

2 Show that if G has ass-cycle, then G has ans − 1-cycle.

We do both and meet in the middle.



A Useful Lemma

Lemma (Gould, Luczak, Pfender, 2003)

Let G be a claw-free graph with minimum degree δ(G ) ≥ 3.

Let C be a cycle of length t with no hops, for some t ≥ 5.

Suppose for some chord xy of C we have V (xCy) has at most 2

vertices whose non-cycle neighbors are not in the cycle.

Then G contains a (t − 1)-cycle.



Build Down

Recall:


If G is a 3-connected claw-free P11-free graph, then G is

hamiltonian.

Since G is 4-connected claw-free P10-free, G is also 3-connectedclaw-free P11-free. Thus G has a hamiltonian cycle.



Build Down

Since G is P10-free, every cycle in G of length 11 or larger has achord.



Build Down

Finding a (t − 1)-cycle from a t-cycle.



Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down




Build Down

A separate argument shows that a cycle of length 9 exists if a cycleof length 10 exists.

Together these give:

Lemma (Ferrara, M, Wenger)

Every 4-connected claw-free P10-free graph G has cycles of length

9 through |V (G )|.



Build Up

We show that there is a 5-cycle and then show that if there is ans-cycle, then there is an (s + 1)-cycle for s = 5, 6, 7.


Every 4-connected claw-free P10-free graph contains cycles of

length 5, 6, 7, and 8.



G has a 3-cycle

Observation

Every 4-connected claw-free graph has a 3-cycle.



Showing G has a 4-cycle.

If G does not have a 4-cycle, then G is 4-regular.




































Recall:

Theorem (Gould, Luczak, Pfender, 1997)

Every 3-connected claw-free P7-free graph is pancyclic.

Thus if G is a 4-connected claw-free P10-free graph that is notpancyclic, then G has an induced P7, P8, or P9.




The case where G has an induced P9.











































































































4-cycle


If G is a 4-connected claw-free P10-free graph, then G contains a

4-cycle or G is the line graph of the Petersen graph.



Putting it all together

Theorem (Ferrara, M, Wenger)

Every 4-connected claw-free P10-free graph is either pancyclic or is

the line graph of the Petersen graph.







Corrollary (Ferrara, M, Wenger)

Every 4-connected claw-free P10-free graph contains cycles of all

lengths except, possibly, 4.







Corrollary (Ferrara, M, Wenger)

Every 4-connected claw-free P10-free graph contains cycles of all

lengths except, possibly, 4.


Every 4-connected claw-free P9-free graph is pancyclic.

Note: This is best possible.Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs


Thank You


Pancyclicity of 4-connected, Claw-free, P10-free Graphs - Morris.pdfPancyclicity How do we show this is true? Build Up 1 Show G has a 3-cycle. 2 Show that if G has an s-cycle, then

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