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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
The Hamiltonian ProblemPancyclicity
Definitions
A hamiltonian cycle is a cycle that contains every vertex
A graph G is hamiltonian if it has a hamiltonian cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Forbidden Subgraphs
Theorem (Goodman, Hedetniemi, 1974)
If G is a 2-connected graph that is claw-free and paw-free, then G
is hamiltonian.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Proof of Goodman, Hedetniemi
Consider a largest cycle and a neighbor not on the cycle.
Claw
Paw
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Proof of Goodman, Hedetniemi
Consider a largest cycle and a neighbor not on the cycle.
Claw
Paw
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Proof of Goodman, Hedetniemi
Consider a largest cycle and a neighbor not on the cycle.
Claw
Paw
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Proof of Goodman, Hedetniemi
Consider a largest cycle and a neighbor not on the cycle.
Claw
Paw
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Proof of Goodman, Hedetniemi
Consider a largest cycle and a neighbor not on the cycle.
Claw
Paw
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Forbidden Subgraphs
Conjecture (Matthews, Sumner, 1984)
Every 4-connected claw-free graph is hamiltonian.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Another Conjecture on 4-connected graphs
Conjecture (Thomassen, 1986)
Every 4-connected line graph is hamiltonian.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Another Conjecture on 4-connected graphs
Conjecture (Thomassen, 1986)
Every 4-connected line graph is hamiltonian.
Theorem (Ryjacek, 1997)
Thomassen’s Conjecture is equivalent to the Matthews-Sumner
Conjecture.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Another Conjecture on 4-connected graphs
Conjecture (Plummer, 1993)
Every 4-connected 4-regular claw-free graph is hamiltonian.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Another Conjecture on 4-connected graphs
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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)
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Forbidden Subgraphs of graphs with higher connectivity
Theorem ( Luczak, Pfender, 2004)
Every 3-connected claw-free P11-free graph is hamiltonian.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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 (Pfender, 2005)
Every 4-connected claw-free -free graph is hamiltonian.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Forbidden Subgraphs of 2-connected graphs
Theorem (Faudree, Gould, 1997)
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 pancyclic if and only if X is the claw
and Y is an induced subgraph of one of the following.
P6N(2, 0, 0)
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Forbidden Subgraphs of 3-connected graphs
Theorem (Gould, Luczak, Pfender, 1997)
Let X and Y be connected graphs such that X ,Y 6= P3 and let G
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)
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Problem (Gould, 2010)
Characterize the pairs of forbidden subgraphs that imply a
4-connected graph is pancyclic.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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)
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Intuition
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Intuition
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
The Goal
Thought (Ferrara, M, Wenger)
Every 4-connected, claw-free, P10-free graph is pancyclic.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Recall:
Theorem ( Luczak, Pfender, 2004)
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Since G is P10-free, every cycle in G of length 11 or larger has achord.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Build Down
Finding a (t − 1)-cycle from a t-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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 )|.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Lemma (Ferrara, M, Wenger)
Every 4-connected claw-free P10-free graph contains cycles of
length 5, 6, 7, and 8.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
G has a 3-cycle
Observation
Every 4-connected claw-free graph has a 3-cycle.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
If G does not have a 4-cycle, then G is 4-regular.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
Showing G has a 4-cycle.
The case where G has an induced P9.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
4-cycle
Lemma (Ferrara, M, Wenger)
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs
The Hamiltonian ProblemPancyclicity
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.
Theorem (Ferrara, M, Wenger)
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
The Hamiltonian ProblemPancyclicity
Thank You
Tim Morris Pancyclicity of 4-connected, Claw-free, P10-free Graphs