Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Bounding the circumference of graphs
Xingxing Yu
School of Mathematics, Georgia Institute of Technology
May 30, 2012
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Tait conjecture (1880): Every 3-connected cubic planar graph
contains a Hamilton cycle.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Tait conjecture (1880): Every 3-connected cubic planar graph
contains a Hamilton cycle.
◮
Tutte gave a counterexample in 1946.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Tait conjecture (1880): Every 3-connected cubic planar graph
contains a Hamilton cycle.
◮
Tutte gave a counterexample in 1946.
◮
Whitney (1931): Every 4-connected planar triangulation
contains a Hamilton cycle.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Tait conjecture (1880): Every 3-connected cubic planar graph
contains a Hamilton cycle.
◮
Tutte gave a counterexample in 1946.
◮
Whitney (1931): Every 4-connected planar triangulation
contains a Hamilton cycle.
◮
Tutte (1956): Every 4-connected planar graph contains a
Hamilton cycle.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Thomassen (1983): Every 4-connected planar graph is
Hamilton-connected.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Thomassen (1983): Every 4-connected planar graph is
Hamilton-connected.
◮
Proof Technique: Tutte path, and Tutte cycles.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Planar graphs
◮
Thomassen (1983): Every 4-connected planar graph is
Hamilton-connected.
◮
Proof Technique: Tutte path, and Tutte cycles.
◮
Chiba and Nishizeki (1989): The Hamiltonian cycle problem is
linear-time solvable for 4-connected planar graphs.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Graphs on projective plane and the Klein bottle
◮
Thomas and Y. (1992): Every 4-connected projective-planar
graph contains a Hamilton cycle.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Graphs on projective plane and the Klein bottle
◮
Thomas and Y. (1992): Every 4-connected projective-planar
graph contains a Hamilton cycle.
◮
Brunet, Nagamoto and Negami (1999): Every 5-connected
triangulation of the Klein bottle is Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Altshuler (1972): 6-Connected toroidal graphs are
Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Altshuler (1972): 6-Connected toroidal graphs are
Hamiltonian.
◮
Brunet and Richter (1995): 5-Connected toroidal
triangulations are Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Altshuler (1972): 6-Connected toroidal graphs are
Hamiltonian.
◮
Brunet and Richter (1995): 5-Connected toroidal
triangulations are Hamiltonian.
◮
Thomas and Y. (1995): Every 5-connected toroidal graph is
Hamiltoinan.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Altshuler (1972): 6-Connected toroidal graphs are
Hamiltonian.
◮
Brunet and Richter (1995): 5-Connected toroidal
triangulations are Hamiltonian.
◮
Thomas and Y. (1995): Every 5-connected toroidal graph is
Hamiltoinan.
◮
Thomas, Y. and Zang (2001): Every 4-connected toroidal
graph contains a Hamilton path.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Nakamoto and Ozeki (2010): Bipartite toroidal
qudrangulations are Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Nakamoto and Ozeki (2010): Bipartite toroidal
qudrangulations are Hamiltonian.
◮
Fujisawa, Nakamoto and Ozeki (2011): 4-Connected toroidal
graphs with toughness exactly 1 are Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Nakamoto and Ozeki (2010): Bipartite toroidal
qudrangulations are Hamiltonian.
◮
Fujisawa, Nakamoto and Ozeki (2011): 4-Connected toroidal
graphs with toughness exactly 1 are Hamiltonian.
◮
Kawarabayashi and Ozeki (2012): 4-Connected toroidal
triangulations are Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Toroidal graphs
◮
Nakamoto and Ozeki (2010): Bipartite toroidal
qudrangulations are Hamiltonian.
◮
Fujisawa, Nakamoto and Ozeki (2011): 4-Connected toroidal
graphs with toughness exactly 1 are Hamiltonian.
◮
Kawarabayashi and Ozeki (2012): 4-Connected toroidal
triangulations are Hamiltonian.
◮
Conjecture (Grünbaum, and Nash-Williams, 1972):
4-Connected toroidal graphs are Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Graphs on other surfaces
◮
Archdeacon, Hartsfield and Little (1992): For each n there is
a triangulation of some surface which is n-connected and has
representativity at least n, and in which every spanning tree
has maximum degree at least n.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Graphs on other surfaces
◮
Archdeacon, Hartsfield and Little (1992): For each n there is
a triangulation of some surface which is n-connected and has
representativity at least n, and in which every spanning tree
has maximum degree at least n.
◮
Y. (1997): 5-Connected “locally planar” triangulations of any
surface are hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Graphs on other surfaces
◮
Archdeacon, Hartsfield and Little (1992): For each n there is
a triangulation of some surface which is n-connected and has
representativity at least n, and in which every spanning tree
has maximum degree at least n.
◮
Y. (1997): 5-Connected “locally planar” triangulations of any
surface are hamiltonian.
◮
Kawarabayashi (2002): Every 5-connected “locally planar”
triangulation is Hamilton connected.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
Projective plane and Klein bottle
Toroidal graphs
Other surfaces
Graphs on other surfaces
◮
Archdeacon, Hartsfield and Little (1992): For each n there is
a triangulation of some surface which is n-connected and has
representativity at least n, and in which every spanning tree
has maximum degree at least n.
◮
Y. (1997): 5-Connected “locally planar” triangulations of any
surface are hamiltonian.
◮
Kawarabayashi (2002): Every 5-connected “locally planar”
triangulation is Hamilton connected.
◮
Conjecture (Zhao; Mohar; Y. 1994). 5-Connected “locally
planar” graphs are Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected planar graphs
◮
Jackson and Wormald (1992): If G is a 3-connected planar
n-vertex graph then circ(G ) = Ω(nc ), where c ≈ 0.2.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected planar graphs
◮
Jackson and Wormald (1992): If G is a 3-connected planar
n-vertex graph then circ(G ) = Ω(nc ), where c ≈ 0.2.
◮
Gao and Y. (1993): Improved c to 0.4.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected planar graphs
◮
Jackson and Wormald (1992): If G is a 3-connected planar
n-vertex graph then circ(G ) = Ω(nc ), where c ≈ 0.2.
◮
Gao and Y. (1993): Improved c to 0.4.
◮
Chen and Y. (2001): Improved c to log3 2 (best possible).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
Two conjectures
◮
Conjecture (Barnette 1969): Every 3-connected bipartite
cubic planar graph has a Hamilton cycle.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
Two conjectures
◮
Conjecture (Barnette 1969): Every 3-connected bipartite
cubic planar graph has a Hamilton cycle.
◮
Conjecture (Bondy 1980?): There is a constant c > 0 such
that every cyclically-4-connected cubic planar n-vertex graph
has a cycle of length cn.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected graphs without K3,t -minors
◮
Chen, Sheppardson, Y. and Zang (2002): If G is a
3-connected n-vertex graph containing no K3,t -minor, then
circ(G ) = Ω(nr (t) ), where r (t) = log8t t+1 (and t ≥ 3).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected graphs without K3,t -minors
◮
Chen, Sheppardson, Y. and Zang (2002): If G is a
3-connected n-vertex graph containing no K3,t -minor, then
circ(G ) = Ω(nr (t) ), where r (t) = log8t t+1 (and t ≥ 3).
◮
Chen, Y. and Zang (2009): If G is a 3-connected n-vertex
graph with no K3,t -minor, then circ(G ) ≥ α(t)nβ , where
α(t) = (1/2)t(t−1) and β = log1729 2.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected graphs without K3,t -minors
◮
Chen, Sheppardson, Y. and Zang (2002): If G is a
3-connected n-vertex graph containing no K3,t -minor, then
circ(G ) = Ω(nr (t) ), where r (t) = log8t t+1 (and t ≥ 3).
◮
Chen, Y. and Zang (2009): If G is a 3-connected n-vertex
graph with no K3,t -minor, then circ(G ) ≥ α(t)nβ , where
α(t) = (1/2)t(t−1) and β = log1729 2.
◮
Question: Could β be replaced with log3 2?
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Planar graphs
3-Connected graphs without K3,t -minors
3-Connected graphs without K3,t -minors
◮
Chen, Sheppardson, Y. and Zang (2002): If G is a
3-connected n-vertex graph containing no K3,t -minor, then
circ(G ) = Ω(nr (t) ), where r (t) = log8t t+1 (and t ≥ 3).
◮
Chen, Y. and Zang (2009): If G is a 3-connected n-vertex
graph with no K3,t -minor, then circ(G ) ≥ α(t)nβ , where
α(t) = (1/2)t(t−1) and β = log1729 2.
◮
Question: Could β be replaced with log3 2?
◮
How about 4-connected graphs without K4,t -minors?
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Graphs with bounded maximum degree
◮
Karger, Motwani, and Ramkumar (1997): Unless P = N P it
is impossible to find (in polynomial time) a cycle of length
V − V ǫ in hamiltonian graphs (for any ǫ < 1).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Graphs with bounded maximum degree
◮
Karger, Motwani, and Ramkumar (1997): Unless P = N P it
is impossible to find (in polynomial time) a cycle of length
V − V ǫ in hamiltonian graphs (for any ǫ < 1).
◮
Conjecture (Karger, Motwani, and Ramkumar 1997): The
same is true for graphs with bounded degree.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Bounded maximum degree
◮
Jackson and Wormald (1993): If G is a 3-connected n-vertex
graph with maximum degree at most d, then
circ(G ) ≥ 12 nr + 1, where r = log6d 2 2.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Bounded maximum degree
◮
Jackson and Wormald (1993): If G is a 3-connected n-vertex
graph with maximum degree at most d, then
circ(G ) ≥ 12 nr + 1, where r = log6d 2 2.
◮
Chen, Xu and Y. (2004): Improved r to log2(d−1)2 +1 2.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Bounded maximum degree
◮
Jackson and Wormald (1993): If G is a 3-connected n-vertex
graph with maximum degree at most d, then
circ(G ) ≥ 12 nr + 1, where r = log6d 2 2.
◮
Chen, Xu and Y. (2004): Improved r to log2(d−1)2 +1 2.
◮
Chen, Gao, Y. and Zang (2006): Further improved r to
max{64, 4d + 1}.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Cubic graphs
◮
Bondy and Simonovits (1980): There is an infinite family of
3-connected cubic n-vertex graphs with circumference Θ(nc ),
where c = log9 8 ≈ 0.946.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Cubic graphs
◮
Bondy and Simonovits (1980): There is an infinite family of
3-connected cubic n-vertex graphs with circumference Θ(nc ),
where c = log9 8 ≈ 0.946.
◮
Jackson (1986): Every 3-connected cubic √
n-vertex graph has
c
circumference Ω(n ), where c = log2 (1 + 5) − 1 ≈ 0.694.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Hardness results
Bounds for graphs with bounded maximum degree
Bounds for cubic graphs
Cubic graphs
◮
Bondy and Simonovits (1980): There is an infinite family of
3-connected cubic n-vertex graphs with circumference Θ(nc ),
where c = log9 8 ≈ 0.946.
◮
Jackson (1986): Every 3-connected cubic √
n-vertex graph has
c
circumference Ω(n ), where c = log2 (1 + 5) − 1 ≈ 0.694.
◮
Bilinski, Jackson, Ma and Y. (2010): If G is a 3-connected
cubic n-vertex graph then the circ(G ) = Ω(|G |0.753 ).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
◮
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Broersma et al (1993): The circumference of a 2-connected
claw-free n-vertex graph is Ω(log n).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
◮
Broersma et al (1993): The circumference of a 2-connected
claw-free n-vertex graph is Ω(log n).
◮
Jackson and Wormald (1993): The circumference of
3-connected K1,d -free graphs, is 12 |G |c . When d = 3 (i.e. G is
claw-free), c = log150 2 ≈ 0.121.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
◮
Broersma et al (1993): The circumference of a 2-connected
claw-free n-vertex graph is Ω(log n).
◮
Jackson and Wormald (1993): The circumference of
3-connected K1,d -free graphs, is 12 |G |c . When d = 3 (i.e. G is
claw-free), c = log150 2 ≈ 0.121.
◮
Bilinski, Jackson, Ma and Y. (2010): If G is a 3-connected
claw-free graph, then circ(G ) ≥ (|G |/12)0.753 + 2.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Ryjáček closure
◮
Let G0 , . . . , Gk be a maximal sequence of graphs such that
G0 = G and for each 1 ≤ i ≤ k, Gi is obtained from Gi −1 by
taking some x ∈ V (G ) for which Gi −1 [NGi −1 (x)] is connected
and adding edges between all pairs of nonadjacent vertices in
NGi −1 (x). Then Gk is said to be a Ryjáček closure of G .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Ryjáček closure
◮
Let G0 , . . . , Gk be a maximal sequence of graphs such that
G0 = G and for each 1 ≤ i ≤ k, Gi is obtained from Gi −1 by
taking some x ∈ V (G ) for which Gi −1 [NGi −1 (x)] is connected
and adding edges between all pairs of nonadjacent vertices in
NGi −1 (x). Then Gk is said to be a Ryjáček closure of G .
◮
Ryjáček (1997): The Ryjáček closure of a claw-free simple
graph G is uniquely determined, and is equal to the line graph
L(H) of a triange-free simple graph H. Furthermore, for every
cycle C ′ of L(H) there exists a cycle C of G with
V (C ′ ) ⊆ V (C ).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
◮
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Ryjáček (1997): The following two conjectures are equivalent.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
◮
Ryjáček (1997): The following two conjectures are equivalent.
◮
Conjecture (Mathews and Sumner 1984): Every 4-connected
claw-free graph is Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
◮
Ryjáček (1997): The following two conjectures are equivalent.
◮
Conjecture (Mathews and Sumner 1984): Every 4-connected
claw-free graph is Hamiltonian.
◮
Conjecture (Thomassen, 1986): Every 4-connected line graph
is Hamiltonian.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Reduction to Eulerian subgraphs
◮
Longest cycle problem on claw-free graphs → Longest cycle
problem on line graphs.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Reduction to Eulerian subgraphs
◮
Longest cycle problem on claw-free graphs → Longest cycle
problem on line graphs.
◮
Longest cycle problem on line graphs → Eulerian subgraph
problem.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Reduction to Eulerian subgraphs
◮
Longest cycle problem on claw-free graphs → Longest cycle
problem on line graphs.
◮
Longest cycle problem on line graphs → Eulerian subgraph
problem.
◮
Bilinski, Jackson, Ma and Y. (2011): Let G be a
3-edge-connected graph, e, f ∈ E (G ), and w : E (G ) → {1, 2}.
Suppose G 6= K23 . Then G contains an Eulerian subgraph H
such that e, f ∈ E (H) and w (H) ≥ (w (G )/6)α + 2, where
α ≈ 0.753 is the real root of 41/x − 31/x = 2.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
We may assume |E (G )| ≥ 7.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
We may assume |E (G )| ≥ 7.
◮
Neither e nor f belongs to a splittable pair in G
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
We may assume |E (G )| ≥ 7.
◮
Neither e nor f belongs to a splittable pair in G
◮
e and f are not adjacent.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
We may assume |E (G )| ≥ 7.
◮
Neither e nor f belongs to a splittable pair in G
◮
e and f are not adjacent.
◮
Neither e nor f is contained in a non-trivial 3-edge-cut of G .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
We may assume |E (G )| ≥ 7.
◮
Neither e nor f belongs to a splittable pair in G
◮
e and f are not adjacent.
◮
Neither e nor f is contained in a non-trivial 3-edge-cut of G .
◮
For any 3-edge-cut S of G , e and f are contained in the same
component of G − S.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
We may assume |E (G )| ≥ 7.
◮
Neither e nor f belongs to a splittable pair in G
◮
e and f are not adjacent.
◮
Neither e nor f is contained in a non-trivial 3-edge-cut of G .
◮
For any 3-edge-cut S of G , e and f are contained in the same
component of G − S.
◮
The vertices incident to e and f all have degree 3 in G .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Need to use splittable pairs of edges (preserving
edge-connectivity).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Need to use splittable pairs of edges (preserving
edge-connectivity).
◮
Use a result of Frank (1992) conerning the existence of
splittable pair at a given vertex.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Need to use splittable pairs of edges (preserving
edge-connectivity).
◮
Use a result of Frank (1992) conerning the existence of
splittable pair at a given vertex.
◮
Chracterize the exceptional cases.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Need to use splittable pairs of edges (preserving
edge-connectivity).
◮
Use a result of Frank (1992) conerning the existence of
splittable pair at a given vertex.
◮
Chracterize the exceptional cases.
◮
Use a result of Jordán which characterize when an edge at a
degree 4 vertex belongs to exactly one splittable pair.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
C1
D1
g1 e h1
g2u v h2
D2
C2
Figure: The structure of G around e.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
C1 , C2 , D1 and D2 are pairwise disjoint, and C1′ , C2′ , D1′ , D2′
are pairwise disjoint.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
C1 , C2 , D1 and D2 are pairwise disjoint, and C1′ , C2′ , D1′ , D2′
are pairwise disjoint.
◮
Let S = {C1 , C2 , D1 , D2 },SS ′ = {C1′ , C2′ , D1′ , D2′ }, and
K = G − {u, v , u ′ , v ′ } − X ∈S∪S ′ X .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
C1 , C2 , D1 and D2 are pairwise disjoint, and C1′ , C2′ , D1′ , D2′
are pairwise disjoint.
◮
Let S = {C1 , C2 , D1 , D2 },SS ′ = {C1′ , C2′ , D1′ , D2′ }, and
K = G − {u, v , u ′ , v ′ } − X ∈S∪S ′ X .
◮
For all X ∈ S and X ′ ∈ S ′ we have either X = X ′ or
X ∩ X ′ = ∅.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
k1
C1
D1
′ g
g
1
v ′ fu ′ 1 ′ g u evhh1
2
g2 2
D2
C2
k2
k1
v′ f
z
k2
D1
e h1
v h2
D2
G∗
G
Figure: C1 = C1′ and C2 = C2′ .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
Y1
Y2
v′
Y1
v′
k1
k1
f
z
v
Y0
e
u
k2
e
g1
g2
C1
C2
g1′
v
Figure: {e, f } is the only splittable pair at z.
Xingxing Yu
Bounding the circumference of graphs
u′
g2′
k2
Y3
f
Y3
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
C1
u′
f
u e
C2
D2
v
v′
K =∅
D1
x1e1
C1 g1′
g1
D2
u ′ g2′
f h2
u e
hv1
g2
′
′
C2 h1v ′ h2 D1
x2 e2
f 2 y2
f 1 y1
K
K 6= ∅
Figure: TheXingxing
structure
of
G when S = S ′ .
Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
k
C1
J1
e1
x1
g1′ u ′ g2′
g1
h2
v
h1
f
u e
g2
y1
C2
h1′ ′ h2′
v
f1
D2
f2
y2
J2
x2
D1
e2
Figure: The case when K has a cut edge separating {x1 , y1 } from {x2 , y2 }
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
◮
◮
◮
Let S ∪ S ′ = {X1 , X2 , . . . , Xq } where
w (X1 ) ≥ w (X2 ) ≥ . . . w (Xq ). Then q ≥ 5.
Xq = C1 , and let r = 41/α − q.
w (K ) ≤ rw (C1 ).
Pq
+ w (K ) ≤ (q − 4 + r )w (Xi ) = (41/α − 4)w (Xi )
for 1 ≤ i ≤ 4.
j=5 w (Xj )
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Choose xi ∈ V (Xi ) for 1 ≤ i ≤ 4.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Choose xi ∈ V (Xi ) for 1 ≤ i ≤ 4.
◮
Suppose that no Eulerian subgraph of G contains
{x1 , x2 , x3 , x4 , e, f }. Then G has a special structure, which
would force S = S ′ , a contradiction.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Choose xi ∈ V (Xi ) for 1 ≤ i ≤ 4.
◮
Suppose that no Eulerian subgraph of G contains
{x1 , x2 , x3 , x4 , e, f }. Then G has a special structure, which
would force S = S ′ , a contradiction.
◮
The cubic case was proved by Ellingham, Holton and Little
(1984).
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Z9
Z5
Z1
e
Z2
Z10
Z1
Z8
f
Z7
Z7
Z4
Z4
Z3
Z3
Z8
f
Z2
Z6
Z5
e
Z6
Figure: No Eulerian subgraph contains e, f and any four given vertices in
Z1 , Z2 , Z3 and Z4 .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Thus {x1 , x2 , x3 , x4 , e, f } is contained in an Eulerian subgraph
H ′ of G . Let G̃ be the graph obtained from G by contracting
Xi to the single vertex xi for 1 ≤ i ≤ 4.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
◮
Thus {x1 , x2 , x3 , x4 , e, f } is contained in an Eulerian subgraph
H ′ of G . Let G̃ be the graph obtained from G by contracting
Xi to the single vertex xi for 1 ≤ i ≤ 4.
◮
We obtain an Eulerian subgraph H̃ of G̃ which contains
{x1 , x2 , x3 , x4 , e, f } from H ′ by contracting the edges in Xi for
1 ≤ i ≤ 4.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Circumference of K1,d -free graphs
Reduction to line graphs
Another conjecture
Outline of proof
Then there is an Eulerian subgraph H of G such that all edges of
H̃ are contained in H and
4
X
(w (Xi )/6)α + w (H̃)
w (H) ≥
i =1

≥ 
4
X
w (Xi ) +
i =1
q
X
j=5
α

α
w (Xj ) + w (K ) /6 + 8
≥ ([w (G ) − 40]/6) + 8
≥ ([w (G ) − 40]/6 + 61/α )α + 2
≥ (w (G )/6)α + 2
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
A new bound
◮
Liu, Y. and Zhang (2011): If G is a 3-connected n-vertex
cubic graph then circ(G ) = Ω(nr ), where r ≈ 0.8 is the real
root of 8.956r + 1.036r = 10.992r .
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
A new bound
◮
Liu, Y. and Zhang (2011): If G is a 3-connected n-vertex
cubic graph then circ(G ) = Ω(nr ), where r ≈ 0.8 is the real
root of 8.956r + 1.036r = 10.992r .
◮
0.8 > log4 3.
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
A new bound
◮
Liu, Y. and Zhang (2011): If G is a 3-connected n-vertex
cubic graph then circ(G ) = Ω(nr ), where r ≈ 0.8 is the real
root of 8.956r + 1.036r = 10.992r .
◮
0.8 > log4 3.
◮
Question: Can r be improved to log5 4?
Xingxing Yu
Bounding the circumference of graphs
Graphs on surfaces
Graphs without K3,t -minor
Graphs with bounded maximum degree
Claw-free graphs
A new bound
Thank you !!!
Xingxing Yu
Bounding the circumference of graphs