Brother trees: A family of optimal 1(p)-hamiltonian and 1-edge hamiltonian graphs

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Brother trees: A family of optimal 1

p

-hamiltonian

and 1-edge hamiltonian graphs

Shin-Shin Kao

a,∗

_{, Lih-Hsing Hsu}

b

a_{Department of Applied Mathematics, Chung-Yuan Christian University, Chong-Li City 320, Taiwan, R.O.C.} b_{Department of Computer and Information Science, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C.}

Received 23 March 2002; received in revised form 2 December 2002 Communicated by F.Y.L. Chin

Abstract

In this paper we propose a family of cubic bipartite planar graphs, brother trees, denoted by BT(n) with n 2. Any BT(n) is hamiltonian. It remains hamiltonian if any edge is deleted. Moreover, it remains hamiltonian when a pair of nodes (one from each partite set) is deleted. These properties are optimal. Furthermore, the number of nodes in BT(n) is 6· 2n− 4 and the diameter is 2n+ 1.

Keywords: Hamiltonian; Diameter; Complete binary tree; Interconnection networks

1. Introduction

An interconnection network connects the proces-sors of the parallel computer. Its architecture can be represented as a graph in which the nodes correspond to the processors and the edges to the communica-tion links. Hence we use graphs and networks inter-changeably. There are many mutually conflicting re-quirements in designing the topology of computer net-works. It is almost impossible to design a network optimum from all aspects. One has to design a suit-able network satisfying the requirements. Diameter is one of the major requirements in designing the topol-ogy of network. Usually a network with smaller

di-* _{Corresponding author.}

E-mail address: [email protected] (S.-S. Kao).

ameter is more preferable. The hamiltonian proper-ties is another requirement. For example, “Token

Pass-ing” approach is used in some distributed operation

systems. Interconnection network requires the pres-ence of hamiltonian cycles in the structure to meet this approach. Fault tolerance is also desirable in mas-sive parallel systems that have a relatively high prob-ability of failure. A number of fault tolerant designs for specific multiprocessor architectures have been proposed based on graph theoretic models in which the processor-to-processor interconnection structure is represented by a graph.

For the graph definition and notation, we follow [1]. G= (V, E) is a graph if V is a finite set and E is a subset of{(a, b) | (a, b) is an unordered pair of V }. We say that V is the node set and E is the edge set of G. Two nodes, a and b, are adjacent if (a, b)∈ E. A path is a sequence of consecutive adjacent nodes. A path

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except for the first node and last node and if they span V . A graph is hamiltonian if it contains a hamiltonian cycle. A graph G= (V, E) is 1-edge hamiltonian if G− e is hamiltonian for any e ∈ E. Obviously, any 1-edge hamiltonian graph is hamiltonian. A 1-edge hamiltonian graph G is optimal if it contains the least number of edges among all 1-edge hamiltonian graphs with the same number of nodes as G. A graph G= (V , E) is 1-node hamiltonian if G− v is hamiltonian for any v ∈ V . A 1-node hamiltonian graph G is

optimal if it contains the least number of edges among

all 1-node hamiltonian graphs with the same number of nodes as G. A graph G= (V, E) is 1-hamiltonian if it is 1-edge hamiltonian and 1-node hamiltonian. A 1-hamiltonian graph G is optimal if it contains the least number of edges among all 1-hamiltonian graphs with the same number of nodes as G. This study of optimal 1-hamiltonian graphs is motivated by optimal fault tolerant token ring design in computer networks. A number of optimal 1-hamiltonian graphs have been proposed [2,5,7]. Obviously, deg_G(x) 3 for any node x in a 1-edge hamiltonian, 1-node hamiltonian, or 1-hamiltonian graph G. It has been proven that any 1-hamiltonian regular graph is optimal if and only if it is 3-regular.

Note that any cycle of a bipartite graph contains the same number of nodes in each partite set. Thus, the deletion of a node from a hamiltonian bipartite graph results in a non-hamiltonian graph. However, the fault tolerant hamiltonian property is not the only factor in designing the topology of networks. For example, the hypercube Qn is a hamiltonian bipartite graph

for n 2. Hence, it is not 1-hamiltonian. When fault occurs, we are interested in the longest cycle in the faulty hypercube [6].

Let G be a bipartite graph with bipartition W and B. We use F(G) to denote {{c, d} | c ∈ W, d ∈ B}. A hamiltonian bipartite graph is 1p-hamiltonian if G − F remains hamiltonian for any F ∈ F(G).

gular torus is (√p ), where p is the number of nodes. In this paper, we propose a family of graphs, called brother trees, denoted by BT(n). The graph BT(n) is planar, bipartite, 3-regular, 1-edge hamiltonian and 1p-hamiltonian. The diameter of the brother tree is (log p), where p is the number of nodes.

2. Definitions and notation

To define brother trees, first we define brother cells. Assume that k is an integer with k 2. The kth brother

cell BC(k) is the five tuple (Gk, wk, xk, yk, zk), where Gk = (V, E) is a bipartite graph with bipartition W

(white) and B (black) and contains four distinct nodes wk, xk, yk and zk. wk is the white terminal; xk the

white root; ykthe black terminal and zkthe black root.

We can recursively define BC(k) as follows:

(1) BC(2) is the 5-tuple (G2, w2, x2, y2, z2) where V (G2) = {w2, x2, y2, z2, s, t}, and E(G2) = {(w2, s), (s, x2), (x2, y2), (y2, t), (t, z2), (w2, z2), (s, t)}.

(2) The kth brother cell BC(k) with k 3 is com-posed of two disjoint copies of (k− 1)th brother cells

BC1(k− 1) =G1_k₋₁, w_k1₋₁, x_k1₋₁, y_k1₋₁, z1_k₋₁,

BC2(k− 1) =G2_k₋₁, w_k2₋₁, x_k2₋₁, y_k2₋₁, z2_k₋₁, a white root xk, and a black root zk. To be specific, V (Gk)= V G1_k₋₁∪ VG2_k₋₁∪ {xk, zk}, E(Gk)= E G1_k₋₁∪ EG2_k₋₁ ∪zk, xk1−1 ,zk, xk2−1 ,xk, z1k−1 , xk, z2k−1 ,y_k1₋₁, w2_k₋₁, wk= w1k−1, and yk= yk2−1.

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Fig. 1. (a) BC(2), (b) BC(3) and (c) BC(4).

Fig. 2. (a) BT(1), (b) BT(3).

BC(2), BC(3), and BC(4) are shown in Fig. 1.

We note that BC1(k− 1) and BC2(k− 1) are iso-morphic for k 3. This property is referred to as the symmetrical property of BC(k). For this reason, we define the degenerate case, BC(1), as the 5-tuple (G1, w1, x1, y1, z1) as V (G1)= {w1, y1}, E(G1)= {(w1, y1)} such that x1= w1and y1= z1.

We can also define the brother cell BC(k) from the complete binary tree B(k), where V (B(k))= {1, 2, . . . , 2k − 1} and E(B(k)) = {(i, j) | j/2 = i}. Assume that k is a positive integer with k 2. The kth brother cell BC(k)= (Gk, wk, xk, yk, zk) can be

constructed by combining two B(k)’s, the upper tree B(k)u and the lower tree B(k)l, and adding edges

between their leaf nodes.

Let n be a positive integer with n 1. The brother

tree, BT(n), is composed of an (n+ 1)th brother

cell BC(n+ 1) = (G1_n₊₁, w1_n₊₁, x_n1₊₁, y_n1₊₁, z1_n₊₁) and an nth brother cell BC(n)= (G2_n, w2_n, x_n2, y_n2, z2_n) with V (G1_n₊₁)∩ V (G2_n)= ∅. To be specific, V (BT(n)) = V (G1_n₊₁)∪V (G2_n) and E(BT(n))= E(G1_n₊₁)∪E(G2_n)

∪ {(z1

n+1, xn2), (yn1+1, w2n), (xn1+1, z2n), (w1n+1, yn2)}.

BT(1) and BT(3) are shown in Fig. 2. Obviously, BT(n) is a 3-regular bipartite planar graph with 6·2n−

4 nodes. Because the (n+ 1)th brother cell is com-posed of two disjoint nth brother cells and two termi-nals, the nth brother tree BT(n) is composed of three disjoint nth brother cells, BC1(n), BC2(n), BC3(n) and two terminals, {x_n1₊₁, z1_n₊₁}. Moreover, BC1(n),

BC2(n) and BC3(n) are arranged in a cyclic order in

BT(n). Thus any two nodes of BT(n) are in the union

of two in the node set of BC1(n), BC2(n), BC3(n) and {x1

n+1, zn1+1}. For this reason, we can assume without

loss of generality that any two nodes of BT(n) are in G1_n₊₁and any edge of BT(n) is in G1_n₊₁. This property is referred to as the symmetrical property of BT(n).

3. Diameter

Theorem 3.1. D(BT(n))= 2n + 1 for any positive

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B(n+ 1)uand B(n+ 1)l. Thus,

(1) both u and v are in V (B(n+ 1)u), or both u and v are in V (B(n+ 1)l), or

(2) u∈ V (B(n + 1)u) and v∈ V (B(n + 1)l).

Now, we introduce some notations before our proof. Let V (B(n+1)u)= {1, 2, . . ., 2n+1−1} and V (B(n+

1)l)= {1, 2, . . . , (2n+1− 1)}. Now, join B(n + 1)u

and B(n + 1)l with the edge set {(2n + i, (2n + i)), ((2n + i), 2n + i + 1) | 0 i 2n − 2} ∪ {(2n+1 _{− 1, (2}n+1 _{− 1)}₎_{} to obtain the brother cell} (G1_n₊₁, w1_n₊₁, x_n1₊₁, y_n1₊₁, z1_n₊₁), where xn+1= 1 and zn+1 = 1 if n is even and xn+1= 1 and zn+1= 1 if

otherwise. Moreover, wn+1= 2nand yn+1= (2n+1−

1).

Case 1: By symmetry, we may assume that both u

and v are in V (B(n+ 1)u). Suppose that u is labeled i and v is labeled j . Obviously, max{log2(i + 1),

log₂(j+ 1)} n + 1. Since dB(n+1)(i, 1)= log2(i+

1) − 1, there exists a path P1 of length log2(i +

1) − 1 joining u to the root of B(n + 1)u. Similarly,

there exists a path P2 of length log2(j + 1) − 1

joining v to 1. Thus,u, P1, 1, P₂−1, v forms a path

joining u to v in B(n+ 1)u. Thus, dBT(n)(u, v) log2(i+ 1) + log2(j+ 1) − 2 2n + 1.

Case 2: We may assume that u is labeled i and v

is labeled j. Without loss of generality, we assume that i j. There then exists a path P1from i to some

leaf node h of B(n+ 1)u of length n− (log2(i+

1) − 1). Let h be a neighborhood of h of BT(n) in B(n+ 1)l. Obviously, there exists a path of length n

from h to the root 1 of B(n+ 1)l. Moreover, there

exists a path P3 of lengthlog2(j+ 1) − 1 joining v to 1. Obviously,u, P1, h, h, P2, 1, P₃−1, v forms

a path joining u to v in G1_n₊₁. Thus, dBT(n)(u, v) 2n+ 1 − log₂(i+ 1) + log₂(j+ 1). Since, i j, dBT(n)(u, v) 2n + 1.

The theorem is proven. ✷

(3) There exists a hamiltonian path P_n3of Gnjoining xnto yn.

(4) There exists a hamiltonian path P4

n of Gnjoining xnto zn.

(5) There exist two disjoint paths P_n5and P_n6such that

(i) they span Gn, (ii) Pn5 joins wn and xn, and

(iii) P_n6joins ynand zn.

(6) There exist two disjoint paths P_n7and P_n8such that

(i) they span Gn, (ii) Pn7 joins wn and zn, and

(iii) P_n8joins xnand yn.

Proof. We prove this lemma by induction. It is easy to check that the lemma holds in BC(2). Assume that the lemma holds for BC(n). By definition, BC(n+ 1) is composed of two disjoint copies of nth brother cells

BC1(n) and BC2(n), a white root xn+1, and a black

root zn+1. By induction,{Pni,1}8_i₌₁exists satisfying the

lemma for BC1(n) and{Pni,2}8i=1 exists satisfying the

lemma for BC2(n). We then set P_n1₊₁=wn+1= wn1, P 5,1 n , x 1 n, zn+1, xn2, (P 5,2 n )−1, w 2 n, y_n1, P_n6,1, z1_n, xn+1, zn2, (P 6,2 n )−1, y 2 n= yn+1 ; P_n2₊₁=wn+1= wn1, Pn2,1, zn1, xn+1, z2n, (P_n4,2)−1, x_n2, zn+1 ; P_n3₊₁=xn+1, zn2, (Pn7,2)−1, wn2, yn1, (Pn3,1)−1, x_n1, zn+1, xn2, Pn8,2, yn2= yn+1 ; P_n4₊₁=xn+1, zn2, (Pn2,2)−1, wn2, yn1, (Pn3,1)−1, x_n1, zn+1 ; P_n5₊₁=wn+1= wn1, Pn1,1, yn1, w2n, Pn7,2, z2n, xn+1 ; P_n6₊₁=yn+1= yn2, (Pn8,2)−1, xn2, zn+1 ; P_n7₊₁=wn+1= wn1, Pn5,1, xn1, zn+1 ; P_n8₊₁=xn+1, zn1, (Pn6,1)−1, yn1, wn2, Pn1,2, yn2= yn+1 . The lemma is proven. ✷

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Lemma 4.2. Assume that BC(n)= (Gn, wn, xn, yn, zn) for some integer n 2. Let e be any edge of BC(n).

Then at least one of the following properties holds.

(1) There exists a hamiltonian path Q1_n(e) of Gn− e

joining wnto yn.

joining wnto zn.

joining xnto yn.

joining xnto zn.

(5) There exist two disjoint paths Q5_n(e) and Q6n(e)

such that (i) they span Gn− e, (ii) Q5n(e) joins wn

and xn, and (iii) Q6n(e) joins ynand zn.

(6) There exist two disjoint paths Q7_n(e) and Q8n(e)

such that (i) they span Gn− e, (ii) Q7n(e) joins wn

and zn, and (iii) Q8n(e) joins xnand yn.

Proof. We prove this lemma by induction. It is easy to check that the lemma holds for BC(2). Assume that the lemma holds for BC(n). By definition, BC(n+ 1) is composed of two disjoint copies of nth brother cells

BC1(n) and BC2(n), a white root xn+1, and a black

root zn+1. Let e be any edge of BC(n+ 1). Using the

symmetrical property of BC(n+ 1), we may assume that e is (zn+1, xn1), (z1n, xn+1), (yn1, w2n), or an edge in

BC1(n). By induction, there exists{Pni,1}8_i₌₁satisfying

the lemma for BC1(n) if e /∈ E(BC1(n)) and there exists {Pni,2}8i=1 satisfying the lemma for BC

2 (n) if e /∈ E(BC2(n)).

Case 1: e= (zn+1, xn1). We set Q7n+1(e) aswn+1= w1_n, Pn1,1, yn1, w2n, P 5,2 n , xn2, zn+1 and Q8_n₊₁(e) as xn+1, z2n, (P 6,2 n )−1, yn2= yn+1.

Case 2: e= (z1_n, xn+1). We set Q5n+1(e) aswn+1= w1_n, Pn1,1, yn1, w2n, P 7,2 n , z2n, xn+1 and Q6_n₊₁(e) as yn+1= yn2, (P 8,2 n )−1, xn2, zn+1.

Case 3: e= (y_n1, w2_n). We set Q_n3₊₁(e) asxn+1, z1n, (Pn4,1)−1, xn1, zn+1, xn2, P

3,2

n , yn2= yn+1.

Case 4: e is in BC1(n). By induction hypothesis, one of the six properties of the lemma holds for

BC1(n). In the following, we find the corresponding paths that satisfy the lemma for BC(n+ 1).

(1) Q7_n₊₁(e)=wn+1= w1n, Q1,1n (e), yn1, wn2, Pn5,2, x_n2, zn+1 ; and Q8_n₊₁(e)=xn+1, z2n, P_n6,2−1, y_n2= yn+1 ; (2) Q2_n₊₁(e)=wn+1= w1n, Q2,1n (e), z1n, xn+1, zn2, P_n4,2−1, x_n2, zn+1 ; (3) Q4_n₊₁(e)=xn+1, z2n, P_n2,2−1, w2_n, y_n1, Q3,1_n (e)−1, x_n1, zn+1 ; (4) Q3_n₊₁(e)=xn+1, z1n, Q_n4,1(e)−1, x1_n, zn+1, xn2, P_n3,2, y_n2= yn+1 ; (5) Q7_n₊₁(e)=wn+1= w1n, Q5,1n (e), xn1, zn+1 ; and Q8_n₊₁(e)=xn₊₁, z1n, Q_n6,1(e)−1, y1_n, w2_n, P_n1,2, y_n2= yn+1 ; and (6) Q2_n₊₁(e)=wn+1= w1n, Q7,1n (e), z1n, xn+1, zn2, P_n2,2−1, w2_n, y_n1,Q8,1_n (e)−1, x_n1, zn+1 . The lemma is proven. ✷

Theorem 4.1. The brother tree BT(n) is 1-edge

hamiltonian for any positive integer n with n 1.

Proof. Note that BT(1) is isomorphic to the hyper-cube Q3. Since Q3 is hamiltonian and edge

sym-metric, BT(1) is 1-edge hamiltonian. Now we con-sider n 2. By definition, BT(n) is composed of an (n+ 1)th brother cell, denoted by BC1(n+ 1) = (G1_n₊₁, w_n1₊₁, x_n1₊₁, y_n1₊₁, z1_n₊₁) and an nth brother cell, denoted by BC2(n)= (G2_n, w2_n, x_n2, y_n2, z2_n). Let e be any edge of BT(n). By the symmetrical property of

BT(n), we assume that e is an edge in G1_n₊₁. Using Lemma 4.1,{Pni,2}8i=1 exists satisfying the lemma for

BC2(n). Using Lemma 4.2, one of the six properties of Lemma 4.2 holds for BC1(n+ 1). In the follow-ing, we find the corresponding hamiltonian cycle He

of BT(n)− e. (1) He= w_n1₊₁, Q1,1_n₊₁(e), y_n1₊₁, w_n2, P_n1,2, y_n2, w1_n₊₁; (2) He= w_n1₊₁, Q2,1_n₊₁(e), z1_n₊₁, x_n2, P_n3,2, y2_n, w1_n₊₁; (3) He= x_n1₊₁, Q3,1_n₊₁(e), y_n1₊₁, w2_n, P_n2,2, z2_n, x1_n₊₁; (4) He= x_n1₊₁, Q4,1_n₊₁(e), z1_n₊₁, x_n2, P_n4,2, z2_n, x_n1₊₁;

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Fig. 3. Illustration of case (6) of Theorem 4.1, where e= (a, x1₄) is the faulty edge. Q7,1₄ is the path joining w₄1to z1₄, Q8,1₄ is the path joining x1₄to y₄1, P₃5,2is the path joining w₃2to x₃2, P₃6,2is the path joining y2₃to z2₃.

(5) He= w1_n₊₁, Q5,1_n₊₁(e), x1_n₊₁, z_n2, (P_n7,2)−1, w_n2, y_n1₊₁, Q6,1_n₊₁(e), z1_n₊₁, x_n2, P_n8,2, y_n2, w1_n₊₁; (6) He= w1_n₊₁, Q7,1_n₊₁(e), z1_n₊₁, x_n2, (P_n5,2)−1, w2_n, y_n1₊₁,Q8,1_n₊₁(e)−1, x_n1₊₁, z2_n, (P_n6,2)−1, y_n2, w1_n₊₁.

The theorem is proven. An illustration is shown in Fig. 3. ✷

5. 1p-hamiltonian

Lemma 5.1. Assume that n is an integer with n 2.

Let BC(n)= (Gn, wn, xn, yn, zn). Suppose that c is

any node of Gn. There then exists a hamiltonian path Rn(c) of Gn− c such that Rn(c) joins ynto zn if c is

a white node, and Rn(c) joins wnto xnif c is a black

node.

Proof. We prove this lemma by induction. It is easy to check that the lemma holds for BC(2). Assume that the lemma holds for BC(n). By definition, BC(n+ 1) is composed of two disjoint copies of nth brother cells BC1(n) = (G1_n, w1_n, x_n1, y_n1, z1_n) and BC2(n) = (G2_n, w_n2, x_n2, y_n2, z2_n), a white root xn+1, and a black

root zn+1. We only prove the case that c is a black

node. Using the symmetrical property of BC(n+ 1),

we may assume that c is a node in BC1(n) or c= zn+1.

Using Lemma 4.1, there exists {Pni,1}8_i₌₁ for BC1(n)

if c /∈ V (BC1(n)) and {Pni,2}8i=1 for BC 2

(n) if c /∈ V (BC2(n)).

Suppose that c is in BC1(n). By induction, there exists a hamiltonian path R1_n(c) of G1_n− c that joins w1

n to xn1. Then, we set Rn+1(c) as wn+1 = wn1, R1_n(c), x_n1, zn+1, xn2, Pn4,2, z2n, xn+1.

Suppose that c= zn+1. We set Rn+1(c) aswn+1= w_n1, Pn1,1, yn1, w2n, P

2,2

n , z2n, xn+1.

The lemma is proven. ✷

Lemma 5.2. Assume that n is an integer with n 2.

Let BC(n)= (Gn, wn, xn, yn, zn). Let c be a white

node of Gnand d be a black node of Gn. Then at least

one of the following properties holds.

(1) There exists a hamiltonian path S_n1(c, d) of Gn− {c, d} joining wnto yn.

(2) There exists a hamiltonian path S_n2(c, d) of Gn− {c, d} joining wnto zn.

(3) There exists a hamiltonian path S_n3(c, d) of Gn− {c, d} joining xnto yn.

(4) There exists a hamiltonian path S_n4(c, d) of Gn− {c, d} joining xnto zn.

(5) There exist two disjoint paths S_n5(c, d) and S_n6(c, d)

such that (i) they span Gn− {c, d}, (ii) Sn5(c, d)

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(6) There exist two disjoint paths S7_n(c, d) and Sn8(c, d)

such that (i) they span Gn− {c, d}, (ii) Sn7(c, d)

joins wnand zn, and (iii) Sn8(c, d) joins xnand yn. Proof. We prove this lemma by induction. It is easy to check that the lemma holds for BC(2). Assume that the lemma holds for BC(n). By definition, BC(n+ 1) is composed of two disjoint copies of nth brother cells BC1(n) and BC2(n), a white root xn+1, and a

black root zn+1. Using the symmetrical property of

BC(n+ 1), we may assume that one of the following

cases holds: (1) c= xn+1and d= zn+1, (2) c= xn+1and d∈ V (BC1(n)), (3) c∈ V (BC1(n)) and d= zn+1, (4) c∈ V (BC1(n)), d∈ V (BC2(n)) or (5) {c, d} ⊂ V (BC1(n)).

In the following, we find the corresponding path(s) for each case.

Case 1: We set S_n1₊₁(c, d) aswn+1= wn1, P 1,1 n , yn1, w2_n, Pn1,2, yn2.

Case 2: With Lemma 5.1, we set S_n1₊₁(c, d) as wn+1= w1n, R1n(d), xn1, zn+1, xn2, P

3,2 n , yn2.

Case 3: With Lemma 5.1, we set S_n3₊₁(c, d) as xn+1, z1n, (R1n(c))−1, yn1, w2n, P

1,2 n , yn2.

Case 4: With Lemma 5.1, we set S_n4₊₁(c, d) as xn+1, z1n, (R1n(c))−1, yn1, w2n, Rn2(d), x2n, zn+1.

Case 5: By induction hypothesis, one of the six

properties of the lemma holds for BC1(n). Note that the endpoints of S_ni(c, d) are the same as the endpoints of Qi_n(e) stated in Lemma 4.2. With a similar argument for the case (4) in Lemma 4.2, we can prove that the lemma is true for this case.

Hence, the lemma is proven. ✷

Theorem 5.1. The brother tree BT(n) is 1p

-hamilto-nian for any positive integer n with n 1.

Proof. This theorem can be obtained with a similar argument of Theorem 4.1. ✷

6. Conclusion

In this paper, we propose a family of bipartite graphs called brother trees, denoted by BT(n). BT(n)

is a planar, bipartite, 3-regular graph, and the number of nodes in BT(n) is 6· 2n− 4. Moreover, we prove that BT(n) is optimal 1-edge hamiltonian and 1p

-hamiltonian, and the diameter of BT(n) is 2n+ 1. Let G be a graph with p nodes and with maximal degree d . The famous Moore bound [2] states that the diameter D(G) is at least log_(d₋₁₎p− 2/d. Thus the diameter of BT(n) is about 2 times the Moore bound. It is interesting to find other optimal 1-edge hamiltonian and 1p-hamiltonian bipartite graphs with

smaller diameters.

We also note that the complete binary tree is one of the most important architectures for interconnection networks [4]. A lot of complete binary tree variations have been proposed. Because the brother tree is com-posed of several complete binary trees, we believe that the brother tree is another candidate for interconnec-tion networks.

Acknowledgements

The authors are very grateful to the anonymous referees for their thorough review of this paper and their concrete and helpful suggestions.

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