mutually independent hamiltonian cycles on cartesian product graphs
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Mutually independent Hamiltonian cycles on Cartesian product graphs
Student: Kai-Siou Wu (吳凱修 ) Adviser: Justie Su-Tzu Juan
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Outline
• Introduction• Motivation & Contribution• Main result– The lower bound of IHC(G1 G2)– IHC(Cm Cn) = 4
• Conclusion & Future work
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Outline
• Introduction• Motivation & Contribution • Main result– The lower bound of IHC(G1 G2)– IHC(Cm Cn) = 4
• Conclusion & Future work
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Introduction
• Hamiltonian cycle– In a graph G, a cycle is called Hamiltonian
cycle if it contains all vertices of G exactly once.
• Independent– Two cycles C1 = u0, u1, ..., un–1, u0 and C2 =
v0, v1, ..., vn–1, v0 in G are independent if u0 = v0 and ui vi for 1 ≤ i ≤ n – 1.
0
4
3 2
1
G
C1 = 0, 1, 2, 3, 4, 0C2 = 0, 2, 3, 4, 1, 0
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Introduction
• Mutually independent– A set of Hamiltonian cycles {C1, C1, …,
Ck} of G are mutually independent if any two dierent Hamiltonian cycles of {C1, C1, …, Ck} are independent.
• Mutually independent Hamiltonicity IHC(G) = k– The mutually independent Hamiltonianicity
of graph G, IHC(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u.
0
4
3 2
1
G
C1 = 0, 1, 2, 3, 4, 0C2 = 0, 2, 3, 4, 1, 0C3 = 0, 3, 4, 1, 2, 0C4 = 0, 4, 1, 2, 3, 0IHC(G) = 4
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Outline
• Introduction• Motivation & Contribution• Main result– The lower bound of IHC(G1 G2)– IHC(Cm Cn) = 4
• Conclusion & Future work
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Motivation & Contribution
• The lower bound of IHC(G1 G2) with different conditions– IHC(G1 G2) IHC(G1)– IHC(G1 G2) IHC(G1) + 2
• Mutually independent Hamiltonianicity of Cm Cn
– IHC(Cm Cn) = 4, for m, n 3.
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Outline
• Introduction• Motivation & Contribution• Main result– The lower bound of IHC(G1 G2)– IHC(Cm Cn) = 4
• Conclusion & Future work
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The lower bound of IHC(G1 G2)
• By definition, we can get the trivial upper bound:IHC(G1 G2) (G1 G2) = (G1) + (G2).
• So, we want to discuss the lower bound of IHC(G1 G2).
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The lower bound of IHC(G1 G2)
Theorem 1. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1 and n 2I1, then IHC(G1 G2) I1.
Theorem 3. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 2, n is even and G1 does not contain C4, then IHC(G1 G2) I1 + 2.
Theorem 2. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 3 , n is odd and m n – 3, then IHC(G1 G2) I1 + 2.Theorem 3. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 2, n is even and G1 does not contain C4, then IHC(G1 G2) I1 + 2.
Theorem 1. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1 and n 2I1, then IHC(G1 G2) I1.
Theorem 2. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 3 , n is odd and m n – 3, then IHC(G1 G2) I1 + 2.
For any Hamiltonian graphs G1 and G2: Let IHC(G1) = I1, |V(G1)| = m, |V(G2)| = n.
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The lower bound of IHC(G1 G2)
• Let x = (u, v) V(G1 G2), where u V(G1) and v V(G2).
G1 G2
(u1, v1)
(u2, v1)
(u1, v2)
(u2, v2)
(u3, v1) (u3, v2) … …G1
G2
H
…
HG1
G1 H
G10 G1 G1 G1
G11
G1n–2
G1n–1
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x1,1
The lower bound of IHC(G1 G2)
• IHC(G1) = I1
• HC1 = e, x1,1, x1,2, x1,3, P1,+,
x1,m–1, x1,m–2, e0
• HC2= e, x2,1, x2,2, x2,3, P2,+,
x2,m–1, x2,m–2, e0• HC
G1
I1
e
x1,2
x1,3
x1,m–1
x1,m–2
P1,+
ex2,m–1
x2,m–2 x2,2
x2,3 P2,+
x2,1
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The lower bound of IHC(G1 G2)
Theorem 1. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1 and n 2I1, then IHC(G1 G2) I1.
Proof.We construct I1 Hamiltonian cycles in G1 G2 starting at vertex e0 first. Then prove these I1 Hamiltonian cycles are mutually independent.
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The lower bound of IHC(G1 G2)
H1
H2
G10 G1
1 G12 G1
3 G14 G1
5 G16
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The lower bound of IHC(G1 G2)
… ……
… …
H3
Hj, 2 j I1
…
G10 G1
1 G12 G1
3 G14 G1
5 G16
2j – 2G1
2j–2 G1n–1
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The lower bound of IHC(G1 G2)
Theorem 1. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1 and n 2I1, then IHC(G1 G2) I1.
Proof.For all 1 j < i I1, we prove that ith Hamiltonian cycle and jth Hamiltonian cycle are independent.
Prove it by induction on i.
For i = 2 is the base case.
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The lower bound of IHC(G1 G2)
66543211 1–
000 0+ 1– 111222 2+ 3– 333444 4+ 5– 555666 6+ 65432100
00122 2+ 23 3– 3344 4+ 45 5– 5566 6+ 10 0+ 00
H1
H2
n 4m 7
x1,10
x2,10
x1,m–10
x2, m–10
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The lower bound of IHC(G1 G2)
Theorem 1. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1 and n 2I1, then IHC(G1 G2) I1.
Proof.
Suppose it is true when i = k – 1 for some 3 k I1.
Now, consider i = k.We show these k Hamiltonian cycles are mutually independent by the following two cases: (1) Hk v.s. H1; (2) Hk v.s. Hj for j {2, 3, …, k – 1};
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The lower bound of IHC(G1 G2)
G1 G2
G1 H
G10 G1
1 G1n–2
G1n–1
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The lower bound of IHC(G1 G2)
Theorem 2. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 3, n is odd and m n – 3, then IHC(G1 G2) I1 + 2.
Proof.We construct I1 + 2 Hamiltonian cycles in G1 G2 starting from vertex e0, and prove these I1 + 2 Hamiltonian cycles are mutually independent. The first I1 Hamiltonian cycles H1 ~ HI1
are the same as those we constructed in Theorem 1.
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The lower bound of IHC(G1 G2)
HI +11
HI +21
G10 G1
1 G12 G1
3 G14 G1
5 G16
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(3) Hk+1 v.s. Hk+2.(2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1};
The lower bound of IHC(G1 G2)
Theorem 2. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 3, n is odd and m n – 3, then IHC(G1 G2) I1 + 2.Proof.Let I1 = k.(1) Hk+1 and Hk+2 v.s. H1;
(3) Hk+1 v.s. Hk+2.
(1) Hk+1 and Hk+2 v.s. H1; (1) Hk+1 and Hk+2 v.s. H1; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (1) Hk+1 and Hk+2 v.s. H1; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (3) Hk+1 v.s. Hk+2.
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The lower bound of IHC(G1 G2)
Theorem 3. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 2, n is even and G1 does not contain C4, then IHC(G1 G2) I1 + 2.
Proof.We construct I1 + 2 Hamiltonian cycles in G1 G2 starting from vertex e0, and prove these I1 + 2 Hamiltonian cycles are mutually independent. The first I1 Hamiltonian cycles H1 ~ HI1
are the same as those we constructed in Theorem 1.
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The lower bound of IHC(G1 G2)
HI +11
HI +21
G10 G1
1 G12 G1
3 G14 G1
5
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The lower bound of IHC(G1 G2)
Theorem 3. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 2, n is even and G1 does not contain C4, then IHC(G1 G2) I1 + 2.Proof.Let I1 = k.
(3) Hk+1 v.s. Hk+2.(2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (1) Hk+1 and Hk+2 v.s. H1;
(3) Hk+1 v.s. Hk+2.
(1) Hk+1 and Hk+2 v.s. H1; (1) Hk+1 and Hk+2 v.s. H1; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (1) Hk+1 and Hk+2 v.s. H1; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (2) Hk+1 and Hk+2 v.s. Hj for j {2, 3, …, I1}; (3) Hk+1 v.s. Hk+2.
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The lower bound of IHC(G1 G2)
Theorem 2. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 3 , n is odd and n – m 4, then IHC(G1 G2) I1 + 2.
Theorem 3. For any Hamiltonian graphs G1 and G2, if m 4I1 – 1, n 2I1 + 2, n is even and G1 does not contain C4, then IHC(G1 G2) I1 + 2.
Corollary 1. For m 7 is odd and n 6 is even, IHC(Cm Cn) = 4.
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Outline
• Introduction• Motivation & Contribution• Main result– The lower bound of IHC(G1 G2)– IHC(Cm Cn) = 4
• Conclusion & Future work
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IHC(Cm Cn) = 4
Corollary 1. For m 7 is odd and n 6 is even, IHC(Cm Cn) = 4.Theorem 4. For n 3 is odd, IHC(Cn Cn) = 4.Theorem 5. For m, n 3 and m, n are odd, IHC(Cm Cn) = 4.Theorem 6. For m 4 is even and n = 3, IHC(Cm Cn) = 4.Theorem 7. For m 4 is even and n = 5, IHC(Cm Cn) = 4.Theorem 8. For m = 4 and n 9 is odd, IHC(Cm Cn) = 4.Theorem 9. For n 4 is even, IHC(Cn Cn) = 4.Theorem 10. For m 6 is even and n = 4, IHC(Cm Cn) = 4.Theorem 11. For m, n 6, IHC(Cm Cn) = 4.
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IHC(Cm Cn) = 43 4 5 6 7 8 9 10 11 12 …
3 …4 …5 …6 …7 …8 …9 …10 …11 …12 …… … … … … … … … … … …
mn
…
IHC(Cm Cn) = 4Corollary 1. m 7 is odd and n 6 is evenTheorem 4. m = n 3 is oddTheorem 5. m, n 3 and m, n are oddTheorem 6. m 4 is even and n = 3Theorem 7. m 4 is even and n = 5Theorem 8. m = 4 and n 7 is oddTheorem 9. m = n 4 is evenTheorem 10. m 6 is even and n = 4Theorem 11. m, n 6
Introduction
• Cm Cn for m, n 3.– 4-regular– C0
– C4
30
(0, 0) (0, 1) (0, 2) (0, 3) (0, 4)
(1, 0) (1, 1) (1, 2) (1, 3) (1, 4)
(2, 0) (2, 1) (2, 2) (2, 3) (2, 4)
(3, 0) (3, 1) (3, 2) (3, 3) (3, 4)
(4, 0) (4, 1) (4, 2) (4, 3) (4, 4)
C6 C5
(5, 0) (5, 1) (5, 2) (5, 3) (5, 4)
C0
C1
C2
C3
C4
C5
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IHC(Cm Cn) = 4
Property 2. For any graph G, if there exist two paths P1, P2 and a subgraph Cn G with Cn = x0, x1, ..., xn–1, x0. And for 0 j n – 1, v – u = a 0 and t1 – t2 = b, P1(t1 + j) = xu+j , P2(t2 + j) = xv+j. If any one of the following conditions holds, then these two paths do not meet the same vertex of Cn at the same time. (1) If a = 0 and b 0. (2) If a 0 and (b – a or n – a).
31
G
Cn
P1
P2
xu
xv
t1
t2
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IHC(Cm Cn) = 4
P1 = x0, x1, ..., xn–1 P2 = x1, x2, ..., xn–1, x0
32
a = 1, (b – a or n – a)
C8
P1
P2
x0t1
t2x1
x2 x3
x4
x5
x6
x7
n = 8t1 – t2 = – 1t1 – t2 = n – 1t1 – t2 = 0
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IHC(Cm Cn) = 4
Property 3. For any graph G, if there exist two paths P1, P2 and a subgraph Cn G with Cn = x0, x1, ..., xn–1, x0. And for 0 j n – 1, v – u = a 0 and t1 – t2 = b, P1(t1 + j) = xu–j , P2(t2 + j) = xv–j. If any one of the following conditions holds, then these two paths do not meet the same vertex of Cn at the same time. (1) If a = 0 and b 0. (2) If a 0 and (b a or a – n).
33
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IHC(Cm Cn) = 4
Theorem 8. For n 7 is odd, IHC(C4 Cn) = 4.
Proof.Since C4 Cn is 4-regular, IHC(C4 Cn) 4.Consider n 9 and without loss of generality we assume that four Hamiltonian cycles starting at e = (0, 0).
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IHC(Cm Cn) = 4
1 2 3
16 15 14
17 18 19
32 31 30
4 5
13 12
20 21
29 28
6 7
11 10
22 23
27 26
8
9
24
25
36
35
34
33
0 1 2 3 4 5 6 7 8
0123
0 1 2 3 4 5 6 7 8
0123
1 34 19
2 33 32
3 4 5
36 35 18
20 21
31 30
6 7
17 16
22 23
29 28
8 9
15 14
24
27
10
13
25
26
11
12
1 36 35
11 10 9
14 15 16
25 24 23
34 33
8 7
17 18
22 21
32 31
6 5
19 29
20 30
3
4
28
27
2
12
13
26
1 11 12
36 26 25
35 27 28
2 10 9
13 14
24 23
29 30
8 7
15 16
22 21
31 32
6 5
17
20
33
4
18
19
34
3
C4 C9
1 2 3
16 15 14
17 18 19
32 31 30
4 5
13 12
20 21
29 28
6 7
11 10
22 23
27 26
8
9
24
25
36
35
34
33
1 34 19
2 33 32
3 4 5
36 35 18
20 21
31 30
6 7
17 16
22 23
29 28
8 9
15 14
24
27
10
13
25
26
11
12
1 36 35
11 10 9
14 15 16
25 24 23
34 33
8 7
17 18
22 21
32 31
6 5
19 29
20 30
3
4
28
27
2
12
13
26
1 11 12
36 26 25
35 27 28
2 10 9
13 14
24 23
29 30
8 7
15 16
22 21
31 32
6 5
17
20
33
4
18
19
34
3
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IHC(Cm Cn) = 4For m = 4, n 9
2+ 1–
0+ 1– 2+ 3– 32100
000 1– 2+ 223 0–
0 3– 0+ 1– 2+ 110
03300+3–01
3–
H3
H4
H1
H2
(0, 1)
(0, 2)
(0, n–1)
(0, n–2)
Property 3
Property 2
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IHC(Cm Cn) = 4
Theorem 8. For n 7 is odd, IHC(C4 Cn) = 4.
Proof.
(1) IHC(C4 Cn) 4.(2) C4 Cn is 4-regular, IHC(C4 Cn) 4.
Hence, IHC(C4 Cn) = 4 can be concluded.
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Outline
• Introduction• Motivation & Contribution• Main result– The lower bound of IHC(G1 G2)– IHC(Cm Cn) = 4
• Conclusion & Future work
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Conclusion
In this thesis, we discuss the lower bound of IHC(G1 G2) and get Theorems 1, 2 and 3.
For any Hamiltonian graphs G1 and G2, let IHC(G1) = I1, |V(G1)| = m, |V(G2)| = n. Theorem 1: If m 4I1 – 1 and n 2I1, then IHC(G1 G2) I1.
Theorem 2: If m 4I1 – 1, n 2I1 + 3, n is odd and m n – 3, then IHC(G1 G2) I1 + 2.Theorem 3: If m 4I1 – 1, n 2I1 + 2, n is even and G1 does not contain C4, then IHC(G1 G2) I1 + 2.
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Conclusion
We also study on IHC(Cm Cn) and get the optimal results as IHC(Cm Cn) = 4 for m, n 3 in Theorems 4, 5, …, and 11.
Corollary 2. For any graphs G1, G2, if G1 and G2 are Hamiltonian and |V(G1)|, |V(G2)| 3, then IHC(G1 G2) 4.
21 lCorollary 3. For graph G = Ck Ck … Ck , l 2, if k1·k2
15, k3 11 and k3, k4, …, kl are odd. For all 3 i l, ki 4(i – 1) + 3 and k1·k2· … · ki–1 ki – 3. Then IHC(G) = 2l.
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Future work
• The exact value of IHC(G1 G2)
• Other interesting graphs
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Thanks for listening.Q & A
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