Download Matthews-Sumner Conjecture and Equivalences

Matthews-Sumner Conjecture and Equivalences
Ralph Faudree
University of Memphis
June 21, 2012
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Forbidden subgraphs have been studied relative to various
Hamiltonian type properties
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Forbidden subgraphs have been studied relative to various
Hamiltonian type properties
Traceable,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Forbidden subgraphs have been studied relative to various
Hamiltonian type properties
Traceable,
Hamiltonian,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Forbidden subgraphs have been studied relative to various
Hamiltonian type properties
Traceable,
Hamiltonian,
Hamiltonian Connected,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Forbidden subgraphs have been studied relative to various
Hamiltonian type properties
Traceable,
Hamiltonian,
Hamiltonian Connected,
Pancyclic,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Forbidden Subgraphs
Definition
A graph G is H-free if G contains no induced copy of the graph H
as a subgraph. More generally, we say G is F-free for some family
of connected graphs F, provided G contains no induced subgraph
isomorphic to a graph in F.
Forbidden subgraphs have been studied relative to various
Hamiltonian type properties
Traceable,
Hamiltonian,
Hamiltonian Connected,
Pancyclic,
Panconnected.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
The Matthews-Sumner Conjecture, 1984
Conjecture A
Every 4-connected claw-free graph is hamiltonian.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Thomassen’s Conjecture, 1986
Conjecture B
Every 4-connected line graph is hamiltonian.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Thomassen’s Conjecture, 1986
Conjecture B
Every 4-connected line graph is hamiltonian.
This conjecture was mentioned as early as 1981
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
4-connected is necessary
Theorem (Jackson, Wormald)
Let G be a 3-connected K1,r -free graph with n vertices. Then G
contains a cycle of length at least nc where
c = (log 26 + 2log 2(2r − 1)) − 1.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
4-connected is necessary
Theorem (Jackson, Wormald)
Let G be a 3-connected K1,r -free graph with n vertices. Then G
contains a cycle of length at least nc where
c = (log 26 + 2log 2(2r − 1)) − 1.
There are infinite families of examples obtained by taking the
inflations (replacing a vertex by a triangle) of appropriate 3-regular
graphs (like the Petersen graph) that imply that the
Jackson-Wormald bound is of the correct order of magnitude (but
not necessarily with the constant c).
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
EIDMA workshop on Hamiltonicity of 2-tough graphs 1996
Line graphs are claw-free, so the Matthews-Sumner Conjecture
implies the Thomassen conjecture.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
EIDMA workshop on Hamiltonicity of 2-tough graphs 1996
Line graphs are claw-free, so the Matthews-Sumner Conjecture
implies the Thomassen conjecture.
Herbert Fleischner conjectured that the two conjectures were
equivalent.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
EIDMA workshop on Hamiltonicity of 2-tough graphs 1996
Line graphs are claw-free, so the Matthews-Sumner Conjecture
implies the Thomassen conjecture.
Herbert Fleischner conjectured that the two conjectures were
equivalent.
This was verified by Zdenek Ryjáček and the result appeared in
1997.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Ryjáček Closure
Theorem (Ryjáček. 1997)
Let G be a claw-free graph. Then
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Ryjáček Closure
Theorem (Ryjáček. 1997)
Let G be a claw-free graph. Then
the closure cl (G ) is uniquely determined,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Ryjáček Closure
Theorem (Ryjáček. 1997)
Let G be a claw-free graph. Then
the closure cl (G ) is uniquely determined,
the circumference of cl (G ) is equal to the circumference of G ,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Ryjáček Closure
Theorem (Ryjáček. 1997)
Let G be a claw-free graph. Then
the closure cl (G ) is uniquely determined,
the circumference of cl (G ) is equal to the circumference of G ,
cl(G) is hamiltonian if and only if G is hamiltonian,
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Ryjáček Closure
Theorem (Ryjáček. 1997)
Let G be a claw-free graph. Then
the closure cl (G ) is uniquely determined,
the circumference of cl (G ) is equal to the circumference of G ,
cl(G) is hamiltonian if and only if G is hamiltonian,
cl(G) is the line graph of a triangle-free graph.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Ryjáček Closure
Theorem (Ryjáček. 1997)
Let G be a claw-free graph. Then
the closure cl (G ) is uniquely determined,
the circumference of cl (G ) is equal to the circumference of G ,
cl(G) is hamiltonian if and only if G is hamiltonian,
cl(G) is the line graph of a triangle-free graph.
This closure was patterned after the Bondy - Chvátal closure
relative to Ore’s Theorem.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Chvátal’s Conjecture, 1973
Toughness Conjecture
There is a positive k such that if t(G ) > k, then G is Hamiltonian.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Chvátal’s Conjecture, 1973
Toughness Conjecture
There is a positive k such that if t(G ) > k, then G is Hamiltonian.
The value of k = 3/2 initially proposed.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Chvátal’s Conjecture, 1973
Toughness Conjecture
There is a positive k such that if t(G ) > k, then G is Hamiltonian.
The value of k = 3/2 initially proposed.
For some time the value k = 2 was investigated, which was
interesting since t(G ) ≥ 2 for any 4-connected claw-free graph.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Chvátal’s Conjecture, 1973
Toughness Conjecture
There is a positive k such that if t(G ) > k, then G is Hamiltonian.
The value of k = 3/2 initially proposed.
For some time the value k = 2 was investigated, which was
interesting since t(G ) ≥ 2 for any 4-connected claw-free graph.
The Chvátal Conjecture is still open.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Status
Theorem (Zhan, 1991)
Every 7-connected line graph is hamiltonian.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Status
Theorem (Zhan, 1991)
Every 7-connected line graph is hamiltonian.
This was proved independently by Bill Jackson, but not published.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Present Status
Theorem (Zhan, 1991)
Every 7-connected line graph is hamiltonian.
This was proved independently by Bill Jackson, but not published.
Theorem (Kaiser and Vrana, 2011)
Every claw-free, 5-connected graph with minimum degree 6 is
hamiltonian connected, and hence hamiltonian.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Hamiltonicity in Line Graphs
Definition
A circuit is a closed walk in a graph that does not repeat edges.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Hamiltonicity in Line Graphs
Definition
A circuit is a closed walk in a graph that does not repeat edges.
Definition
A dominating circuit is a circuit in which each edge of the graph is
either on the circuit, or incident to a vertex on the circuit.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Hamiltonicity in Line Graphs
Definition
A circuit is a closed walk in a graph that does not repeat edges.
Definition
A dominating circuit is a circuit in which each edge of the graph is
either on the circuit, or incident to a vertex on the circuit.
Theorem (Harary and Nash-Williams, 1965)
Let H be a graph with at least 3 edges. Then L(H) is hamiltonian
if and only if H = K1,r with r ≥ 3 or H contains a dominating
circuit.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Appropriate Edge-Connectivity for 4-connected Line Graph
Question
What is the counterpart of 4-connectivity in L(H)?
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Appropriate Edge-Connectivity for 4-connected Line Graph
Question
What is the counterpart of 4-connectivity in L(H)?
Definition
A graph H is essentially 4-edge connected if it contains no
edge-cut R such that |R| < 4 and at least two components of
H − R contain an edge.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
The Dominating Circuit Conjecture
Theorem
L(H) is 4-connected if and only if H is essentially 4-edge connected.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
The Dominating Circuit Conjecture
Theorem
L(H) is 4-connected if and only if H is essentially 4-edge connected.
Conjecture C
Every essentially 4-edge connected graph contains a dominating
circuit.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
4-edge Connectivity is a Stronger Condition
Note that 4-edge connected graphs contain two edge-disjoint
spanning trees.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
4-edge Connectivity is a Stronger Condition
Note that 4-edge connected graphs contain two edge-disjoint
spanning trees.
Hence, 4-edge connected graphs contain dominating circuits.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
4-edge Connectivity is a Stronger Condition
Note that 4-edge connected graphs contain two edge-disjoint
spanning trees.
Hence, 4-edge connected graphs contain dominating circuits.
Hence, line graphs of 4-edge connected graphs are
hamiltonian.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Cubic Graphs
Definition
A graph H is cyclically 4-edge-connected if H contains no edge-cut
R such that |R| < 4 and at least 2 components of H − R contain a
cycle.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Cubic Graphs
Definition
A graph H is cyclically 4-edge-connected if H contains no edge-cut
R such that |R| < 4 and at least 2 components of H − R contain a
cycle.
Remark
A cubic graph is essentially 4-edge-connected if and only if it is
cyclically 4-edge-connected.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Ash and Jackson Cubic Graph Conjecture
Conjecture D
Every cyclically 4-edge-connected cubic graph has a dominating
cycle.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Ash and Jackson Cubic Graph Conjecture
Conjecture D
Every cyclically 4-edge-connected cubic graph has a dominating
cycle.
Theorem (Fleischner and B. Jackson 1989)
Conjecture D is equivalent to conjectures A, B, and C .
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Fleischner’s Conjecture
Conjecture E
Every cyclically 4-edge-connected cubic graph that is not
3-edge-colorable has a dominating cycle.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Fleischner’s Conjecture
Conjecture E
Every cyclically 4-edge-connected cubic graph that is not
3-edge-colorable has a dominating cycle.
Theorem (Kochol 2000)
Conjecture E is equivalent to Conjectures A, B, C , and D.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Snark Conjecture
Definition
A snark is a cyclically 4-edge-connected cubic graph of girth at
least 5 that is not 3-edge-colorable.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Snark Conjecture
Definition
A snark is a cyclically 4-edge-connected cubic graph of girth at
least 5 that is not 3-edge-colorable.
Conjecture F
Every snark has a dominating cycle.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Snark Conjecture
Definition
A snark is a cyclically 4-edge-connected cubic graph of girth at
least 5 that is not 3-edge-colorable.
Conjecture F
Every snark has a dominating cycle.
Theorem (Broersma, Fijavz, Kaiser, Kužel, Ryjáček and Vrana,
2008)
Conjectures A − F are equivalent.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Stronger Appearing Equivalent Conjectures
Conjecture G
Every 4-connected claw-free graph is hamiltonian connected.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Stronger Appearing Equivalent Conjectures
Conjecture G
Every 4-connected claw-free graph is hamiltonian connected.
Conjecture H
Every 4-connected line graph is hamiltonian connected.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Stronger Appearing Equivalent Conjectures
Conjecture G
Every 4-connected claw-free graph is hamiltonian connected.
Conjecture H
Every 4-connected line graph is hamiltonian connected.
Theorem (Kužel, Ryjáček, Vána, Xiong)
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Stronger Appearing Equivalent Conjectures
Conjecture G
Every 4-connected claw-free graph is hamiltonian connected.
Conjecture H
Every 4-connected line graph is hamiltonian connected.
Theorem (Kužel, Ryjáček, Vána, Xiong)
1
Conjectures G and H are equivalent.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Stronger Appearing Equivalent Conjectures
Conjecture G
Every 4-connected claw-free graph is hamiltonian connected.
Conjecture H
Every 4-connected line graph is hamiltonian connected.
Theorem (Kužel, Ryjáček, Vána, Xiong)
1
Conjectures G and H are equivalent.
2
Conjectures A − H are equivalent.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
1-Hamiltonian-Connected
Definition
A graph G is 1-Hamilton-connected if for any vertex x of G there
is a Hamilton path in G − x between any two vertices.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
1-Hamiltonian-Connected
Definition
A graph G is 1-Hamilton-connected if for any vertex x of G there
is a Hamilton path in G − x between any two vertices.
Conjecture I
Every 4-connected claw-free graph is 1-Hamiltonian Connected.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
1-Hamiltonian-Connected
Definition
A graph G is 1-Hamilton-connected if for any vertex x of G there
is a Hamilton path in G − x between any two vertices.
Conjecture I
Every 4-connected claw-free graph is 1-Hamiltonian Connected.
Theorem (Ryjáček, Saburov, and Vána, 2011)
Conjectures A − I are equivalent.
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Survey Paper
How Many Conjectures can you Stand? a Survey
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences
Survey Paper
How Many Conjectures can you Stand? a Survey
by Broersma Ryjáček, and P. Vrána
Ralph Faudree
Matthews-Sumner Conjecture and Equivalences