Final_Report_96_(Generalized honeycomb torus is Hamiltonian)

8/14/2019 Final_Report_96_(Generalized honeycomb torus is Hamiltonian)

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Xiaofan Yang , David J. Evans ,Hongjian Lai , GrahamM. Megson

Information Processing Letters 92

(2004) 3137

As presented by Ying-Jhih Chen

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A (di)graph is called hamiltonian if itcontains Hamilton cycle.

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Let n be a positive even integer, mbe a positive integer, and dbe anonnegative integerthat is less than

n and is of the same parity as m. An(m,n, d) generalized honeycombtorus, denotedby GHT(m,n, d), is a

graph with vertex set V( GHT(m, n, d) )={ : i { 0, 1,

, m1 }, j { 0, 1, , n1} }.

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Note: Here and in what follows, all arithmetic operationscarried out on the first and second components aremodulo m and n, respectively.

Two vertices and with i k are adjacentif

and only if one of the following three conditions issatisfied:

(a) = or = ;

(b) 0 i m2, i+j is odd, and = ;

(c) i = 0, j is even, and = .

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Let p and q be two positive integers.Let g(p, q) denote the smallestpositive integer s satisfying p s = 0

(mod q). Then g(p, q) = q / gcd(p,q) .

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p ( q / gcd(p, q) ) = ( p / gcd(p, q) ) q 0 (modq)

Let s be an integer with 1 s ( q / gcd(p,q) ) 1.

Ifp s = 0 (mod q)

Since gcd( p / gcd(p,q) , q / gcd(p,q) ) = 1

g(p, q) q /gcd(p,q)

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g(p, q) q /gcd(p,q)

r N s.t.p s =r q.( p/gcd(p,q) ) s = r ( q /

gcd(p,q) )

q / gcd(p,q) | s ( 1 s ( q / gcd(p,q) )

1 )

g(p, q) = q / gcd(p,q)


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Given two positive integers a and b,we need to consider a graph G(a, b)that has {0, 1, . . .,a 1} as the

vertex set and { < i, i + b >: 0 i a 1} as the edge set, where thearithmetic is modulo a.

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If gcd(a, b) = 1, then G(a, b) is acycle (loop and multiple edgesinclusive).

e. g. G( 3, 2)

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Consider the infinite sequence (0, b,2b, 3b, . . .) of neighboring vertices.

By Lemma 1

(0, b, 2b, 3b, . . ., (a 1)b, 0)forms a cycle.

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.

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If m is even, then set {P(k gcd(n,d), gcd(n, d)): 0 k ( n / gcd(n,d) ) 1} constitutes a path

decomposition of GHT(m,n, d). (Wecall this path decomposition asstandard path decomposition.)

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GHT(4, 12, 4):

P(0, 4)

P(4, 4)

P(8, 4)


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If m is even. Then GHT(m,n, d) isHamiltonian.

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We construct a graph G( n /gcd(n,d) , d/gcd(n,d) )

By Lemma 2, the sequence of neighboring

vertices ( 0, d/gcd(n, d), 2 d/gcd(n, d), . . . ,((n /

gcd(n, d)) 1 ) d/gcd(n, d), 0 )

forms a cycle.

This cycle can be extended to aHamiltonian cycle of GHT(m,n, d) accordingto the following steps:

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Step 1: For each i with 0 i (n /gcd(n,d)) 1, let

V (i)= { : 0 p m1, i gcd(n, d) q (i + 1) gcd(n, d) 1.

Step 2: Replace each vertex i of G( n

/gcd(n,d) , d/gcd(n,d) ) with V (i).

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Step 3: Replace each edge (i, i + d)of G( n /gcd(n,d) , d/gcd(n,d) ) witha path of GHT(m,n, d) obtained from

path P(igcd(n, d), gcd(n, d)) byadding the following edge:

(, ).

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P(0, 4)

P(4, 4)

P(8, 4)

GHT(4, 12 ,4)


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If m is odd, then GHT(m,n, d) is Hamiltonian. Poof: We construct a graph G( n /gcd(n,d+1) , d+1

/gcd(n,d+1) ) in the way given in Lemma 2.

By Lemma 2, the sequence

( 0, (d + 1)/gcd(n, d + 1), 2 ((d +1)/gcd(n, d + 1)),. . . , (n/gcd(n, d + 1) 1) ((d + 1)/gcd(n, d + 1)),0 )

Forms a cycle.This cycle can be extended to a

Hamiltonian cycle of GHT(m,n, d) according to thesome steps.

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GHT (5, 12, 5 )

P(6, 2), P(8, 4)

P(0, 2), P(2, 4)


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Every generalized honeycomb torusis Hamiltonian.

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Final_Report_96_(Generalized honeycomb torus is Hamiltonian)