Optimal 1-Hamiltonian graphs

ELSEVIER Information Processing Letters 65 (1998) 157-161

Optimal 1 -hamiltonian graphs

Jeng-Jung Wang, Chun-Nan Hung, Lih-Hsing Hsu *

Department of Computer and Information Science, National Chiao Tung University Hsinchu 30050, Taiwan, ROC Received 10 June 1997

Communicated by T. Asano

Abstract

In this paper, we present a family of 3-regular, planar, and hamiltonian graphs. Any graph in this family remains hamiltonian if any node or any edge is deleted. Moreover, the diameter of any graph in this family is O(G) where p is the number of nodes. @ 1998 Elsevier Science B.V.

Keywords: Hamiltonian cycle; Token ring; Fault tolerance

1. Introduction

An interconnection network connects the processors of the parallel computer. Its architecture can be rep- resented as a graph in which the nodes correspond to the processors and the edges to the communication links. Hence, we use graph and network interchangeably. There are lot of mutually conflicting requirements in designing the topology of computer networks. It is almost impossible to design a network which is optimum from all aspects. One has to design a suitable network depending on the requirements and their properties. The hamiltonian properties is one of the major requirements in designing the topology of network. For example,

“Token Passing” approach is used in some distributed operation systems. Interconnection network requires

the presence of hamiltonian cycles in the structure to meet this approach. Fault tolerant is also desirable in massive parallel systems that have a relatively high probability 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.

In this paper, a network is represented as an undirected graph. For the graph theoretical definition and notation we follow [ 11. G = (YE) is a graph if V is a finite set and E is a subset of {(a, b) 1 (a, b) is an unordered pair of V}. We say that V is the node set and E is the edge set of G. Let p = IV1 and q = 1 El. Two nodes a

and b are adjacent if (a, 6) E E. A path is a sequence of nodes such that two consecutive nodes are adjacent. A path is delimited by (x0, xi, x2,. . .,n,_l). WeuseP-’ todenotethepath (.~~-i,...,n~,x~,xc) ifP is the path (x0,x1,x2,..., x,-l). A path is called a hamiltonian path if its nodes are distinct and they span V. A

* Corresponding author. Email: [email protected].

’ This work was supported in part by the National Science Council of the Republic of China under contract NSC86-2213-E009-020.

0020-0190/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PIISOO20-0190(98)00004-O

the graph in the family is lp/6J + 2 if p is even and b/S] + 3 if p is odd. Similar problem has been discussed by Harary and Hayes [2,3]. A graph G is k-node hamiltonian if G - V’ is hamiltonian for any V’ c V with IV’1 = k, and a graph G is k-edge hamiltonian if G - E’ is hamiltonian for any E’ c E with IE’I = k. A p-node

k-node hamiltonian graph is optimal if its number of edges is the smallest, and a p-node k-edge hamiltonian graph is optimal if its number of edges is the smallest. In [ 31, Harary and Hayes presented a family of optimal p-node k-node hamiltonian graphs for every positive integer k, and in whereas [ 21, Harary and Hayes presented a family of optimal p node k-edge hamiltonian graphs for every positive integer k. For the special case k = 1, the family of optimal l-node hamiltonian graphs proposed in [3] is actually the family of optimal l-edge hamiltonian graphs proposed in [2]. Hence this family of graphs is 1-hamiltonian. Any single graph in this family is planar and of diameter [(p + 1) /3j.

In [3], Harary and Hayes were not sure that their proposed optimal l-node hamiltonian graphs are of only such optimal graphs and asked for the determination of all such graphs. With the family of graphs proposed by Mukhopadhyaya and Sinha mentioned above, we can easily solve Harary and Hayes’ problem with a counterexample. Note that a l-node hamiltonian graph may not be necessarily hamiltonian. A graph is

hypohumi~toniun if G is not hamiltonian but G - u is hamiltonian for every u E V. There are a lot of studies on

hypohamiltonian graphs [ 4,6,8]. It seems that the problem of determination of all optimal l-node hamiltonian graphs is very difficult.

In this paper, we propose a family of optimal 1-hamiltonian graphs. Each graph in this family is planar and of diameter 0( fi). The degree of each node in any graph of this family is 3.

2. Definitions

In this section, we are going to present a family of graphs G(k) for every positive integer k. There are 2k + 1 disjoint isomorphic induced subgraphs, G( k, i) with 0 < i < 2k, in G(k). For 0 6 i 6 2k, G( k, i) is the graph

(V( k, i), E( k, i)) where

V(k, i) = {Xi’,j 1 1 < j < k} U {x:,~ / 1 < i < k} U {Y;, Zi}, and E(k,i)={(xf,j,xf,j_1) 11 <j < k}‘J{($,jvx;j+l) 11 <j< k}

U{(X~,~,X;J> 11 ~j~kk)U{(X:,,Yi),(~i,~i),(~i,X~,,)}. In Fig. 1, we illustrate the graph G( k, i) .

Then, we define the graph G(k) = (V(k) , E(k)) with

V(k)= U V(k,i), and

O<i<2k

E(k)= IJ E(k,i) U {(zi,ti+Imti(2t+l)) IO G i G 2k}U{(x:,,x~++lmod(2k+l),k) IO G i G 2k)-

J.-J. Wang et al. /Information Processing Letters 65 (1998) 157-161 159

Fig. I.

Fig. 2.

In Fig. 2, we illustrate the graph G( 2).

Obviously, the graph G(k) is planar and with p = 4k2 + 6k + 2 nodes. Each vertex of G(k) is of degree 3. Assume that x is in V( k, i) and y is in V( k, j) . It is easy to see that the distance between x and z; is at most

k + 1, the distance between z; and z,i is at most k and the distance between z,i and y is at most k + 1. Hence

the diameter of G(k) is 0( J7r).

3. Hamiltonian properties

Lemma 1. G( k, i) has a hamiltonian path joining zi with x:,~ for every positive integer k, i with 0 < i < 2k. Proof. Suppose that k is an odd integer.

there is also a hamiltonian path joining Zi with ~f,~. We use P(i, z, I) to denote the hamiltonian path joining ti with xfk. We also use Q(i, r, I) to denote the path < x~,~,x:,~_, , . . . , XI,,, yi, xf,,, x$, . . . ,x$ >. With these notationq’it is easy to see that

ko

---4k

/

form a hamiltonian cycle of G(k). By the symmetric property of G(k), it is easy to see that G(k) is l-edge fault tolerant.

To discuss the l-node fault tolerant property of G(k), we need the following lemma.

Lemma 2. For any positive integer j with 1 6 j < k, G( k, i) - (~1,~) h as a hamiltonian path joining zi with xfk or xik.

Proof. We first assume that j is an odd integer and k is an even integer. It is easy to see that

(Zi, J’i, Xi,t 9 XL.1 9 Xi,29 Xf,29 Xf,j, * . * 3 Xf,,i-l* Xf,j, Xf,j+l, X:,j+l* $,j+2t . . . v Xf,kt x;k)

is a hamiltonian path joining zi with xc,k. Other cases can be similarly discussed. Hence the lemma is proved. 0

We may assume the faulty node of G(k) is ya, ZO, or ~6,~ for 1 < j < k. Assume the fault node is zo. It is easy to see that

(x&k QUbA I P-‘(l,z,r) - XO,k, x;,k A Zl % Z2 -x:k P(2,zJ) , ’ x;,k P-‘(3,z,r) -23,..., &-l,k P-‘(2i-l,z,r) P(2i.z.O

p Z2i- 1 T Z2i -x!2ik , ,..., Z2k - Q(2k,r4) X2k,kv x;,k) 1

form a hamiltonian cycle in G(k) - { ZO}.

Assume the faulty node is ye. It can be checked that

(,$,X;),k_ ,,..., .&,&,& ,..., Xb,k,n;,,:~.:,k,x;,k~x:,k ,...,

X;k-2,k ____f Q(2k-2,rd X2k-2,k l 9 x;k- 1,k

P-‘(Zk-1,z.r)

- Z2k-1, Z2k-2,. . . , ZO, Z2k - Q(2k,r,O X2k,k, x;),k) 1

form a hamiltonian cycle in G(k) - {yo}

Finally, we assume that the faulty node is ~6,~ for 1 < j 6 k. It follows from Lemma 2 and the symmetric property of G(k), we may assume that there exists a hamiltonian path R joining ze with xb,k in G( k, 0) - (~6,~). Suppose that k is even. Then

J.-J. Wang et al. /Information Processing Letters 65 (1998) 157-161 161 Q(~J-& 1 QCb.0 I

-Xl,k~X;,k-x2k ) ,...,

x;k- l,k - Q(2k-l,r,0 x2k- 1 ,k’ dk,k [ ~Z2k,Z2k-l,Z2k-l,...,Z1,ZO) P-‘(2k,z,r)

is a hamiltonian cycle.

Similarly, we can discuss the case that k is odd. Thus G(k) is l-node fault tolerant. Since the degree of any node in G(k) is 3, G(k) is an optimal 1-hamiltonian graph.

Theorem 3. G(k) is an optimal 1-hamiltonian graph with diameter 0( ,/jT) where p is the number of nodes

in G(k).

References

[I] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North-Holland, New York, 1980. [2] F. Harary, J.P. Hayes, Edge fault tolerance in graphs, Networks 23 (1993) 135-142. [ 31 F. Harary, J.P. Hayes, Node fault tolerance in graphs, Networks 27 (1996) 19-23.

[4] P. Ho& J. Sir&i, On a construction of Thomassen, Graphs and Combinatorics 2 (1986) 347-350.

[5] K. Mukhopadhyaya, B.P. Sinha, Hamiltonian graphs with minimum number of edges for fault-tolerant topologies, Inform. Process. Lett. 44 (1992) 95-99.

[6] L. Stacho, Maximally non-hamiltonian graphs of girth 7, Graphs and Combinatorics 12 ( 1996) 361-371.

[7] T.Y. Sung, T.Y. Ho, L.H. Hsu, Optimal k-fault-tolerant networks for token rings, J. Inform. Sci. and Engineering (to appear). [8] C. Thomassen, Hypohamiltonian and hypotraceable graphs, Discrete Math. 9 ( 1974) 91-96.