University of Illinois, Urbana 2013
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Hamiltonian Cycles in the Square of a Graph
Jan EKSTEIN
[email protected]
Název prezentace
Department of Mathematics, University of West Bohemia, Pilsen
Prednášejíci
CE-ITI - Center of Excellence - Institut for Theoretical Computer
Science, Prague
NTIS - New Technologies for the Information Society, Pilsen
Power of the graph
G n ....... n-th power of G : V (G n ) = V (G ) NTIS – NEW TECHNOLOGIES FOR INFORMATI
.
edges between vertices at distance ≤ n in G
Název prezentace
Prednášejíci
Power of the graph
G n ....... n-th power of G : V (G n ) = V (G ) NTIS – NEW TECHNOLOGIES FOR INFORMATI
.
edges between vertices at distance ≤ n in G
Theorem
[Sekanina] Let G be a connected graph. Then G 3 is hamiltonian
connected.
Název prezentace
Prednášejíci
Power of the graph
G n ....... n-th power of G : V (G n ) = V (G ) NTIS – NEW TECHNOLOGIES FOR INFORMATI
.
edges between vertices at distance ≤ n in G
Theorem
[Sekanina] Let G be a connected graph. Then G 3 is hamiltonian
connected.
Název prezentace
•..
..
S(K1,3 ) :
•...
Prednášejíci
..
....
....•....
...... .............
.
.
.
.
.
•......
...•
.......
......
•....
•.
Square of the graph
Theorem
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[Fleischner] Let G be a 2-connected graph. Then G 2 is
hamiltonian.
Název prezentace
Prednášejíci
Square of the graph
Theorem
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[Fleischner] Let G be a 2-connected graph. Then G 2 is
hamiltonian.
Theorem
[El Kadi Abderrezzak, Flandrin, Ryjáček] IfNázev
G is a prezentace
connected
graph such that every induced S(K1,3 ) has at least three edges in a
block of degree at most 2, then G 2 is hamiltonian.Prednášejíci
Square of the graph
Theorem
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[Fleischner] Let G be a 2-connected graph. Then G 2 is
hamiltonian.
Theorem
[El Kadi Abderrezzak, Flandrin, Ryjáček] IfNázev
G is a prezentace
connected
graph such that every induced S(K1,3 ) has at least three edges in a
block of degree at most 2, then G 2 is hamiltonian.Prednášejíci
Theorem
[Thomassen] If the block graph of G is a path, then G 2 is
hamiltonian.
Connection of the graphs
Theorem
– NEW TECHNOLOGIES FOR INFORMATI
[Fleischner] Let y and z be arbitrarily chosenNTIS
vertices
of a
2
2-connected graph G . Then G contains a hamiltonian cycle C
such that the edges of C incident with y are in G and at least one
of edges of C incident with z is in G . If y and z are adjacent in G ,
then these are three different edges.
Název prezentace
Prednášejíci
Connection of the graphs
Theorem
– NEW TECHNOLOGIES FOR INFORMATI
[Fleischner] Let y and z be arbitrarily chosenNTIS
vertices
of a
2
2-connected graph G . Then G contains a hamiltonian cycle C
such that the edges of C incident with y are in G and at least one
of edges of C incident with z is in G . If y and z are adjacent in G ,
then these are three different edges.
Název prezentace
G = G1 [x1 = x2 ]G2
G1
G2
......................... .............
.
.....
..... .....
....
x1 •..... x2 ....
.
...
.
... ....... ......
......
..................... .....
Prednášejíci
G ... connected graph; G 2 is hamiltonian and x ∈ V (G )
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Název prezentace
Prednášejíci
G ... connected graph; G 2 is hamiltonian and x ∈ V (G )
a) Vertex x is of type 1: ... ∃ C in G 2 ; both
edges C
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containing x are edges in G .
Název prezentace
Prednášejíci
G ... connected graph; G 2 is hamiltonian and x ∈ V (G )
a) Vertex x is of type 1: ... ∃ C in G 2 ; both
edges C
NTIS – NEW TECHNOLOGIES FOR INFORMATI
containing x are edges in G .
b) Vertex x is of type 2: ... x is not of type 1 and ∃ C in G 2 ;
exactly one edge C containing x is edge in G .
Název prezentace
Prednášejíci
G ... connected graph; G 2 is hamiltonian and x ∈ V (G )
a) Vertex x is of type 1: ... ∃ C in G 2 ; both
edges C
NTIS – NEW TECHNOLOGIES FOR INFORMATI
containing x are edges in G .
b) Vertex x is of type 2: ... x is not of type 1 and ∃ C in G 2 ;
exactly one edge C containing x is edge in G .
c) Vertex x is of type 3: ... x is not of type 1 or 2 and ∃ C in
G 2 ; there exist u, v ∈ NG (x) such that uv ∈ E (C ).
Název prezentace
Prednášejíci
G ... connected graph; G 2 is hamiltonian and x ∈ V (G )
a) Vertex x is of type 1: ... ∃ C in G 2 ; both
edges C
NTIS – NEW TECHNOLOGIES FOR INFORMATI
containing x are edges in G .
b) Vertex x is of type 2: ... x is not of type 1 and ∃ C in G 2 ;
exactly one edge C containing x is edge in G .
c) Vertex x is of type 3: ... x is not of type 1 or 2 and ∃ C in
G 2 ; there exist u, v ∈ NG (x) such that uv ∈ E (C ).
Název prezentace
d) Vertex x is of type 4: ... x is not of type 1 or 2 or 3.
Prednášejíci
G ... connected graph; G 2 is hamiltonian and x ∈ V (G )
a) Vertex x is of type 1: ... ∃ C in G 2 ; both
edges C
NTIS – NEW TECHNOLOGIES FOR INFORMATI
containing x are edges in G .
b) Vertex x is of type 2: ... x is not of type 1 and ∃ C in G 2 ;
exactly one edge C containing x is edge in G .
c) Vertex x is of type 3: ... x is not of type 1 or 2 and ∃ C in
G 2 ; there exist u, v ∈ NG (x) such that uv ∈ E (C ).
Název prezentace
d) Vertex x is of type 4: ... x is not of type 1 or 2 or 3.
Prednášejíci
V[i] (G ) = {x ∈ V (G ) | x is of type i}, i = 1, 2, 3, 4.
Theorem
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Let G1 , G2 be connected graphs such that (G1 )2 , (G2 )2 are
hamiltonian, let xi ∈ V (Gi ), i = 1, 2. If
S
I) G = G1 [x1 = x2 ]G2 and xi ∈ V[1] (Gi )
V[2] (Gi ),
i = 1, 2, or
II) G = G1 [x1 = xS
2 ]K2 ,
S
x1 ∈ V[1] (G1 )
V[2] (G1 )
V[3] (G1 )Název
and prezentace
V (K2 ) = {x2 , u} or
III) G = G1 [x1 = x2 ]G2 , x1 ∈ V[3] (G1 ) and x2Prednášejíci
∈ V[1] (G2 ),
2
then G is hamiltonian.
Moreover under the assumptions of I),
a) if xi ∈ V[1] (Gi ), i = 1, 2, then x = x1 = x2 ∈ V[1] (G );
NTIS – NEW TECHNOLOGIES FOR INFORMATI
b) if x1 ∈ V[1] (G1 ) and x2 ∈ V[2] (G2 ), then
x = x1 = x2 ∈ V[2] (G );
S
c) if G2 is 2-connected and x1 ∈ V[1] (G1 ) V[2] (G1 ), then
v ∈ V[1] (G ) for any v ∈ V (G2 ), v 6= x2 ;
d) if xi ∈ V[2] (Gi ), i = 1, 2,
S then
Název prezentace
x = x1 = x2 ∈
/ V[1] (G ) V[2] (G ).
Moreover under the assumptions of II),
a) if x1 ∈ V[1] (G1 ), then x = x1 = x2 ∈ V[1] (GPrednášejíci
);
b) if x1 ∈ V[2] (G1 ),Sthen x = x1 = x2 ∈ V[2] (G );
c) if x1 ∈ V[1] (G1 ) V[2] (G1 ), then u ∈ V[2] (G ).
Bl(G ) → block graph of G : V (Bl(G )) = cut vertices, blocks
NEW TECHNOLOGIES FOR INFORMATI
.
→ edge between cut vertex a and blockNTIS
B – of
G : a ∈ V (B)
Název prezentace
Prednášejíci
Bl(G ) → block graph of G : V (Bl(G )) = cut vertices, blocks
NEW TECHNOLOGIES FOR INFORMATI
.
→ edge between cut vertex a and blockNTIS
B – of
G : a ∈ V (B)
Theorem
Let G be a graph such that its block graph is a path and let u be
an arbitrary vertex which is not a cut vertex andNázev
is contained
in an
prezentace
endblock of G . Then G 2 is hamiltonian and moreover
a) if u is contained in a cyclic block, then u ∈ V[1] (G ),
b) if u is contained in an acyclic block, then uPrednášejíci
∈ V[2] (G ).
Main results
V≥3 (G ) = {x ∈ V (G )|dG (x) ≥ 3}
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Název prezentace
Prednášejíci
Main results
V≥3 (G ) = {x ∈ V (G )|dG (x) ≥ 3}
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- for x ∈ V (G ), tG (x) denotes the number of acyclic non-end
blocks of G containing x
Název prezentace
Prednášejíci
Main results
V≥3 (G ) = {x ∈ V (G )|dG (x) ≥ 3}
NTIS – NEW TECHNOLOGIES FOR INFORMATI
- for x ∈ V (G ), tG (x) denotes the number of acyclic non-end
blocks of G containing x
Lemma
Názevinprezentace
Let G be a connected graph with exactly one vertex
V≥3 (Bl(G )) corresponding to a cut vertex a of G . If a is contained
2 contains a
in at most two acyclic non-end blocks of G , then GPrednášejíci
hamiltonian cycle C such that
if tG (a) = 0, then both edges of C incident with a are in G ,
if tG (a) = 1, then exactly one edge of C incident with a is in G ,
if tG (a) = 2, then no edge of C incident with a is in G .
Theorem
Let G be a connected graph with at least three vertices such that
NTIS – NEW TECHNOLOGIES FOR INFORMATI
i) every vertex x ∈ V≥3 (Bl(G )) corresponds to a cut vertex of
G , and
ii) for any two vertices x, y ∈ V≥3 (Bl(G )), distBl(G ) (x, y ) ≥ 4.
Then G 2 is hamiltonian if and only if every cut Název
vertex of
G is
prezentace
contained in at most two acyclic non-end blocks of G .
Prednášejíci
Theorem
Let G be a connected graph with at least three vertices such that
NTIS – NEW TECHNOLOGIES FOR INFORMATI
i) every vertex x ∈ V≥3 (Bl(G )) corresponds to a cut vertex of
G , and
ii) for any two vertices x, y ∈ V≥3 (Bl(G )), distBl(G ) (x, y ) ≥ 4.
Then G 2 is hamiltonian if and only if every cut Název
vertex of
G is
prezentace
contained in at most two acyclic non-end blocks of G .
Prednášejíci
The proof of Theorem is made by induction and Lemma is
necessary in the induction hypothesis. The conditions in the
theorem can be verified in polynomial time.
Corollary
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Let G be a connected graph such that its block graph Bl(G ) is
homeomorphic to a star in which the center corresponds to a cut
vertex a of G . Then the graph G 2 is hamiltonian if and only if the
vertex a is contained in at most two acyclic non-end blocks of G .
Název prezentace
Prednášejíci
Corollary
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Let G be a connected graph such that its block graph Bl(G ) is
homeomorphic to a star in which the center corresponds to a cut
vertex a of G . Then the graph G 2 is hamiltonian if and only if the
vertex a is contained in at most two acyclic non-end blocks of G .
Název prezentace
Corollary
If the block graph of G with at least three verticesPrednášejíci
is a star, then
G 2 is hamiltonian.
Graphs with block graph homeomorphic with a star
Denote Bc a block corresponding to a center of a star
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Název prezentace
Prednášejíci
Graphs with block graph homeomorphic with a star
Denote Bc a block corresponding to a center of a star
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Acceptable cycle in G :
Název prezentace
Prednášejíci
Graphs with block graph homeomorphic with a star
Denote Bc a block corresponding to a center of a star
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Acceptable cycle in G :
- hamiltonian cycle C in (Bc )2
Název prezentace
Prednášejíci
Graphs with block graph homeomorphic with a star
Denote Bc a block corresponding to a center of a star
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Acceptable cycle in G :
- hamiltonian cycle C in (Bc )2
- for each cut vertex vi ∈ V (Bc ) ∃ edge vi wi ∈ E (C ) such that
vi wi ∈ E (Bc ), vi wi are pairwise distinct
Název prezentace
Prednášejíci
Graphs with block graph homeomorphic with a star
Denote Bc a block corresponding to a center of a star
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Acceptable cycle in G :
- hamiltonian cycle C in (Bc )2
- for each cut vertex vi ∈ V (Bc ) ∃ edge vi wi ∈ E (C ) such that
vi wi ∈ E (Bc ), vi wi are pairwise distinct
Název prezentace
Theorem
Prednášejíci
Let G be a connected graph such that its block graph Bl(G ) is
homeomorphic to a star in which the center corresponds to a block
Bc of G . If (Bc )2 contains an acceptable cycle in G , then G 2 is
hamiltonian.
Corollary
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Let G be a connected graph such that its block graph Bl(G ) is
homeomorphic to a star in which the center corresponds to a block
Bc of G . If Bc is hamiltonian, then G 2 is hamiltonian.
Název prezentace
Prednášejíci
Corollary
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Let G be a connected graph such that its block graph Bl(G ) is
homeomorphic to a star in which the center corresponds to a block
Bc of G . If Bc is hamiltonian, then G 2 is hamiltonian.
Theorem
Název prezentace
[Schaar] For every block G with |V (G )| ≥ 4 it holds that there
Prednášejíci
exists a hamiltonian cycle in G 2 such that it contains
at least four
edges of G .
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Conjecture:
Let G be a connected graph such that its block
graph Bl(G ) is homeomorphic to a star in which the center c
corresponds to a block Bc of G . If dBl(G ) c ≤ k, k < 5, then G 2 is
hamiltonian.
Název prezentace
Prednášejíci
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Conjecture:
Let G be a connected graph such that its block
graph Bl(G ) is homeomorphic to a star in which the center c
corresponds to a block Bc of G . If dBl(G ) c ≤ k, k < 5, then G 2 is
hamiltonian.
Název prezentace
Conjecture is true for k ≤ 2 but for k = 3, 4 is an open problem.
Prednášejíci
..........
....
.•....
... .....
..K....n...2....
........ ..........
.. ..
.. ....•....... ....
... ..... ...... .....
.
.. .... Kn3 .... ...
........................•.............................
.. .
.
•.............. .......... .........•..
.
... .....
.
.
... ...... ....... ...
... ..... ..... ..
... ....•....... .
... ... .. ..
... ......... ..
...Kn .
... .4.
... ..
.
.....•........
Kn5 .... ....
....
Kn1 ...... ......
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Název prezentace
Prednášejíci
Conclusion
NTIS – NEW TECHNOLOGIES FOR INFORMATI
Thank you for your attention.
Název prezentace
This contribution was supported by the European Regional
Prednášejíci for
Development Fund (ERDF), project NTIS - New Technologies
the Information Society, European Centre of Excellence,
CZ.1.05/1.1.00/02.0090.