The Balanced Hamiltonian Cycle Problem

The Balanced Hamiltonian Cycle Problem Student: Hou-Ren Wang Advisor: Dr. Justie Su-Tzu Juan Date: 2011/07/28

Outline • Introduction • Preliminary • Main Result • The Balanced Hamiltonian Cycle in Hypercube • The Balanced Hamiltonian Cycle in Torus Graphs • Conclusion &Future Work

Introduction • The bit conversion of gray-code usually concentrates on some parts. • If we can average the bit conversion into all different bits, then there will be more efficient. • The questions on bit conversion of gray-code can take as the questions of balanced Hamiltonian cycles in hypercube.

Preliminary • Hypercube • G = Qn = (V, E) • V = {x1x2…xn| xi{0, 1}, i = 1, 2, …, n} • E = {xy | x, y V, Σ (xi yi)= 1} • Dimension • n is the dimension of Qn n i=1

Preliminary • The i-th dimensional edge • {xy | xy E(C), xi yi= 1, xk= yk, k  i } • Ei(C)= the set of i-th dimensional edges in Hamiltonian cycle.

Preliminary • The decomposition of hypercube 011 111 101 001 010 110 Q3 000 100

Preliminary • Torus Graphs T(n, m) = Cm Cn • An example of T(7, 3) = C3 C7 • 1-dimensional edge is the edges of horizontal dimension. • 2-dimensional edge is the edges of vertical dimension.

Preliminary • Balanced Hamiltonian cycle • Ei(C)= the set of i-th dimensional edges in Hamiltonian cycle. • Balanced Hamiltonian cycle is a Hamiltonian cycle satisfied that: | | Ei(C) |  |Ej(C)||  1, for i j. | E1(C) | = 2 | E2(C) | = 2

Main Result • The Balanced Hamiltonian Cycle in Hypercube Qn • Case of n  2k • Case of n= 2, 4 and 8 • The Balanced Hamiltonian Cycle in Torus Graphs • T(n, n) • T(mn, n) • T(m, 3)

The Balanced Hamiltonian Cycle in Hypercube • Case 1. Prove that there is no balanced Hamiltonian cycle in Qn for n 2k • Lemma 1 • Lemma 2 • Theorem 1 • Case 2. Find a balanced Hamiltonian cycle in Qn for n = 2, 4 and 8 • Algorithm 1

Case 1 Lemma 1. For each set Ei(C) that contains all i-th dimensional edges of Balanced Hamiltonian Cycle C in Qn, | Ei(C)| is even. Proof. (1/2) Suppose there exists a Balanced Hamiltonian Cycle C in Qn such that | Ei(C) | is odd for 1  i  n. DevideQn into two components and that without i-th dimensional edges.

Proof. (2/2) Then the origin vertex and the terminate vertex of C would not be the same. There is a contradiction. So | Ei(C) | is even for 1  i  n. Origin vertex … Terminate vertex

Lemma 2. For each set Ei(C) that contains all i-thdimen-sional edges of Balanced Hamiltonian Cycle C,  2n/n   | Ei(C) |  2n/n . Proof. (1/2) | E1(C) | | E2(C) | | Em(C) | … | Ei(C) | = l + 1, for 1  i  m. | Em+ 1(C) | | Em + 2(C) | | En(C) | … | Ej(C) | = l, for m + 1  j  n.

Proof. (2/2)   Because | | Ei(C) |  |Ej(C)||  1, for i j.   2n/n   | Ei(C) |  2n/n

Theorem 1. There is no Balanced Hamiltonian Cycle in Qn when n 2k for any positive integer k. Proof. Suppose there exists a Balanced Hamiltonian Cycle in Qn for n 2k. | Ei(C) | = 2n/n and | Ej(C) | =2n/n for 1  i  m < j  n. (Lemma 2) | El (C) | is odd for some 1  l  n. ()(Lemma 1)

Case 2 A balanced Hamiltonian cycle for hypercube Q2. A balanced Hamiltonian cycle for hypercube Q4.

Algorithm 1 This algorithm finds a Balanced Hamiltonian Cycle in Q8. Step 1.

Step 2.

Step 3.Let , for 0 i 15, and shown as figure below.

Step 4.

Step 5.

The Balanced Hamiltonian Cycle in Torus Graphs • Algorithm 2 • For T(n, n), n is a positive integer, n 3. • Algorithm 3 • For T(mn, n), n is an odd positive integer, m is a positive integer, m, n 3. • Algorithm 4 • For T(mn, n), n is an even positive integer, m is a positive integer, m, n 3. • Algorithm 5 • For T(m, 3), m is a positive integer, m 3.

Algorithm 2 | E1(C*) | = n2 / 2 | E2(C*) | = n2 / 2 | E1(C*) | = 13 | E2(C*) | = 12 | E1(C*) | = 8 | E2(C*) | = 8 | E1(C*) | = 5 | E2(C*) | = 4

Algorithm 3 • Use the program below to reduce the Hamiltonian path P1. inti, j; for (i = 0; i < n; i ++) for (j = 0; j < n; j ++) { P1 = P1( i, (i j)(mod n) ) ; }

The Hamiltonian path P1 for T(5, 5). | E1(P1) | = n  1 = 4 | E2(P1) | = (n  1)  n = 20

The Hamiltonian path P2 for T(5, 5).

The Hamiltonian path P2 for T(5, 5). | E1(P2) | = (n  1)  n  1 =19 | E2(P2) | = n = 5

The Hamiltonian path P for T(5, 5). P occurs from step 3 of Algorithm 2. | E1(P) | = n2 / 2 = 12 | E1(P ) | = n2 / 2 = 12 | E2(P ) | = n2 / 2 = 12 | E2(P) | = n2/ 2 = 12

If m is an odd positive integer P2 P1 P2 P1 • P … | E1(C*) | = n2 ((m 1) / 2) + (n2 + 1) / 2 | E2(C*) | = n2 ((m 1) / 2) + (n2  1) / 2

If m is an even positive integer P2 P1 … P2 P1 | E1(C*) | = n2 (m / 2) | E2(C*) | = n2 (m / 2)

Algorithm 4 • Case of an even integer n • P • P • P | E1(C*) | = mn2/ 2 | E2(C*) | = mn2/ 2

Algorithm 5 • The Hamiltonian path R1 for T(4, 3). | E1(R1) | = 5 | E2(R1) | = 6

The Hamiltonian path R2 for T(5, 3). | E1(R2) | = 6 | E2(R2) | = 8

Define k = m / 3  1, we can discuss in to four cases • When m 1 (mod 3) and k is odd. • When m 1 (mod 3) and k is even. • When m 2 (mod 3) and k is odd. • When m 2 (mod 3) and k is even.

when m 1 (mod 3) and k is odd. • P2 • P1 • P • R1 | E1(C*) | = 9  (( m / 3  2) / 2 + 11 | E2(C*) | = 9  (( m / 3  2) / 2 + 10

when m 1 (mod 3) and k is even. • P2 • P1 • P1 • R1 | E1(C*) | = 9  (( m / 3  2) / 2 + 6 | E2(C*) | = 9  (( m / 3  2) / 2 + 6

when m 2 (mod 3) and k is odd. • R2 • P2 • P1 • P | E1(C*) | = 9  (( m / 3  2) / 2 + 12 | E2(C*) | = 9  (( m / 3  2) / 2 + 12

when m 2 (mod 3) and k is even. • P2 • P1 • P1 • R2 | E1(C*) | = 9  (( m / 3  2) / 2) + 7 | E2(C*) | = 9  (( m / 3  2) / 2) + 8

Conclusion & Future Work • This thesis proves that there is no Balanced Hamiltonian Cycle in Qnwhen n 2k. • This work also proposes a scheme to find a Balanced Hamiltonian Cycle in Q8. Conjecture 1. For any positive integer k 4, there is a balanced Hamiltonian cycle in Qn, for n = 2k.

Conclusion & Future Work • In torus graph, we find the balanced Hamiltonian cycle of T(n, n), T(mn, n) and T(n, 3) for any positive integer n 3 and m. • Future work can try to find the balanced Hamiltonian cycle on T(n, 4), T(n, 5), …, T(n, m) for m, n 3. • We can also try to find Balanced Hamiltonian Cycles in some other kinds of graphs, just like Star Graph, Generalize Hypercube …, etc.

The Balanced Hamiltonian Cycle Problem