Construction Schemes for Fault-Tolerant Hamiltonian Graphs

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1 Constuction Schemes fo Fault-Toleant Hamiltonian Gaphs Jeng-Jung Wang Depatment of Compute and Infomation Science, National Chiao Tung Univesity, Hsinchu, Taiwan 30050, Republic of China Chun-Nan Hung Depatment of Compute and Infomation Science, National Chiao Tung Univesity, Hsinchu, Taiwan 30050, Republic of China Jimmy J. M. Tan Depatment of Compute and Infomation Science, National Chiao Tung Univesity, Hsinchu, Taiwan 30050, Republic of China Lih-Hsing Hsu Depatment of Compute and Infomation Science, National Chiao Tung Univesity, Hsinchu, Taiwan 30050, Republic of China Ting-Yi Sung Institute of Infomation Science, Academia Sinica, Taipei, Taiwan 115, Republic of China In this pape, we pesent thee constuction schemes fo fault-toleant Hamiltonian gaphs. We show that applying these constuction schemes on fault-toleant Hamiltonian gaphs geneates gaphs peseving the oiginal Hamiltonicity popety. We apply these constuction schemes to geneate some known families of optimal 1-Hamiltonian gaphs in the liteatue and the Hamiltonicity popeties of these gaphs ae the diect consequence of the constuction schemes. In addition, we can use these constuction schemes to popose new family of optimal 1- Hamiltonian gaphs John Wiley & Sons, Inc. Keywods: Hamiltonian gaphs; token ings; fault toleance; 1-Hamiltonian gaphs; 3-joins; cycle extensions 1. INTRODUCTION The topology of an inteconnection netwok fo pa- Received Decembe 1998; accepted Novembe 1999 Coespondence to: T.-Y. Sung; Contact gant sponso: National Science Council of the Republic of China, contact gant numbe NSC M c 2000 John Wiley & Sons, Inc. allel and distibuted systems can always be epesented by a gaph. Let G =(V, E) be an undiected gaph. A cycle in G that taveses evey vetex exactly once is called a Hamiltonian cycle. A gaph G is called a Hamiltonian gaph o said to be Hamiltonian if it contains a Hamiltonian cycle. In this pape, we study Hamiltonian gaphs which coespond to the token ing topology. Fault toleance is an impotant issue in the design of an inteconnection netwok. When faults occu in a netwok, it coesponds to emoving edges and/o vetices fom the gaph. Let V V and E E. WeuseG V to denote the subgaph of G induced by V V, and G E, the subgaph obtained by emoving E fom G. Faults can be in the combination of vetices and edges. Let F V E. We use G F to denote the subgaph induced by V F and deleting the edges in F fom the induced subgaph. If G V is Hamiltonian fo any V V and V = k, then G is called a k-vetex-hamiltonian gaph. If G E is Hamiltonian fo any E E and E = k, then G is called a k-edge-hamiltonian gaph. If G F is Hamiltonian fo any F V E and F = k, then G is called a k-hamiltonian gaph. In this pape, we ae, in paticula, inteested in k = 1 and an abitay NETWORKS, Vol. 35(3),

2 vetex o edge fault, that is, a 1-Hamiltonian gaph. Fo convenience, we wite G f instead of G {f}, whee {f} V E. A k-hamiltonian gaph is said to be optimal if it contains the least numbe of edges among all k-hamiltonian gaphs having the same numbe of vetices. Mukhopadhyaya and Sinha [4], Haay and Hayes [1,2], Wang et al. [5], Hung et al. [3], and Wang et al. [6] poposed diffeent families of optimal 1-Hamiltonian gaphs. These optimal 1-Hamiltonian gaphs ae all tivalent except the ones poposed in [1,2,4] with an odd numbe of vetices which have exactly one vetex of degee 4 and the emaining vetices of degee 3. In this pape, we popose two opeations on gaphs called 3-join and cycle extension. In addition, a vaiation of 3-join, called (3,4)-join, is also pesented. Using these opeations on paticula gaphs, the esultant gaphs have nice popeties. In paticula, applying these opeations on tivalent 1-Hamiltonian gaphs can yield othe tivalent 1-Hamiltonian gaphs. Futhemoe, by ecusively applying these opeations on specific simple pimitive gaphs, we can obtain the optimal 1-Hamiltonian gaphs poposed in [1 6]. Ou constucted gaphs can be easily shown to be optimal 1-Hamiltonian using the popeties of these opeations. This pape is oganized as follows: In Section 2, we intoduce the 3-join opeation and its vaiation (3,4)-join and thei popeties. In addition, we show that optimal 1- Hamiltonian gaphs in [1 5] can be constucted by ecusively applying 3-joins and (3,4)-joins on specific pimitive gaphs. In Section 3, the cycle extension opeation is pesented. We also study its popeties and show that the gaphs poposed in [6] can be obtained by ecusively applying cycle extensions. In addition, we apply this opeation on the Petesen gaph (which is 1-vetex- Hamiltonian) to geneate a new family of 1-Hamiltonian gaphs. Final emaks appea in Section JOIN In this section, we intoduce two opeations, called 3- join and (3,4)-join, pefomed on two diffeent gaphs. A 3-join is an opeation applied on two vetices of degee 3 in two gaphs, while a (3,4)-join is applied on a vetex of degee 3 in one gaph and a vetex of degee 4 in anothe gaph. We will show that pefoming these two opeations on two 1-Hamiltonian gaphs poduces new families of 1-Hamiltonian gaphs. Futhemoe, those families of 1-Hamiltonian gaphs poposed in [1 5] can be easily geneated by applying the two opeations on specific gaphs and K 4. In the following discussion, we use 3-join and (3,4)-join to mean the gaphs and the opeations intechangeably Definitions and Popeties Let G 1 and G 2 be two gaphs. We assume that V(G 1 ) V(G 2 )= thoughout the section. Let x be a vetex of a gaph. We use N(x) to denote an odeed set which consists of all of the neighbos of x, that is, N(x) isan odeing of all of the neighbos of x. Hencefoth, we use N(x) as an odeed set. The 3-join, as illustated in Figue 1, is defined as follows: Definition 1. Let x be a vetex of degee 3 in G 1 and y be a vetex of degee 3 in G 2. Let N(x) ={x 1,x 2,x 3 } and N(y) ={y 1,y 2,y 3 }. The 3-join of G 1 and G 2 at x and y is a gaph K given by V(K) = (V(G 1 ) {x}) (V(G 2 ) {y}) E(K) = (E(G 1 ) {(x, x i ) 1 i 3}) (E(G 2 ) {(y, y i ) 1 i 3}) {(x i,y i ) 1 i 3}. and Note that diffeent N(x) and N(y) geneate diffeent 3-joins of G 1 and G 2 at x and y. Fo example, as illustated in Figue 1, given N(y) ={y 1,y 2,y 3 }, the 3-join of G 1 and G 2 at x and y with N(x) ={x 1,x 2,x 3 } is diffeent fom the one with N(x) ={x 2,x 1,x 3 }. On the othe hand, each 3-join of G 1 and G 2 at x and y is uniquely detemined by N(x) and N(y). Thoughout this subsection, we use x to epesent a vetex of degee 3 in gaph G 1, and y, a vetex of degee 3 in gaph G 2. A gaph K is said to be a 3-join of G 1 and G 2 if K is a 3-join of G 1 and G 2 at x and y with some N(x) and some N(y). Clealy, 3-joins of two tivalent gaphs G 1 and G 2 ae still tivalent. In this pape, we wite a path P fom x to y as x, x 1,x 2,...,x k,y o x P y. Theoem 1. Let G 1 and G 2 be two gaphs and K be a 3-join of G 1 and G 2. If both G 1 and G 2 ae 1-Hamiltonian gaphs, then K is a 1-Hamiltonian gaph. Poof. Let K be a 3-join of G 1 and G 2 at x and y with N(x) ={x 1,x 2,x 3 } and N(y) ={y 1,y 2,y 3 }. Let f be any fault, vetex o edge, of K. Conside f (x i,y i ) fo all 1 i 3. Without loss of geneality, we may assume that f (V(G 1 ) {x}) (E(G 1 ) {(x, x i ) 1 i 3}). Since G 1 is 1-Hamiltonian, it follows that thee is a Hamiltonian cycle H 1 in G 1 f given by x, x i P x j,x, whee i, j {1, 2, 3} with i j and P denotes a path in G 1. Let k be the unique FIG. 1. Examples of 3-joins of two tivalent gaphs. 234 NETWORKS 2000

3 element in {1, 2, 3} {i, j}. Since G 2 is 1-Hamiltonian, thee is a Hamiltonian cycle H 2 in G 2 (y, y k ) which can be witten as y, y i Q y j,y. Hence, x i P x j,y j Q y i,x i foms a Hamiltonian cycle of K f. Conside f =(x i,y i ) fo some i. Without loss of geneality, we may assume that f =(x 1,y 1 ). Let H 1 be a Hamiltonian cycle of G 1 (x, x 1 ) which can be witten as x, x 2 P x 3,x. Let H 2 be a Hamiltonian cycle of G 2 (y, y 1 ) which can be witten as y, y 2 Q y 3,y. Then, x 2 P x 3,y 3 Q y 2,x 2 foms a Hamiltonian cycle of K f. Hence, the theoem is poved. Using a simila poof technique in Theoem 1, we can show the following coollaies: Coollay 1. Let G 1 and G 2 be two gaphs and K be a 3-join of G 1 and G 2. If both G 1 and G 2 ae 1-edge- Hamiltonian, then K is also 1-edge-Hamiltonian. We conside a special 3-join on G 1 and G 2, whee G 2 is K 4, as descibed below: Definition 2. Let G be a gaph with a vetex v of degee 3. A 3-vetex expansion on v is a gaph obtained fom G by pefoming a 3-join at v and any vetex of K 4. In othe wods, a 3-vetex expansion on v of degee 3 is a gaph obtained fom G by eplacing the vetex v with a K 3 and connecting the thee vetices of K 3 to the thee neighbos of v, one by one. Note that 3-vetex expansions on v ae unique up to isomophism. Coollay 2. Let G be a tivalent 1-edge-Hamiltonian gaph and V (G) ={v V(G) G v is not Hamiltonian}. Let G denote the gaph obtained fom G by pefoming a sequence of 3-vetex expansions on evey vetex v V (G). Then, G is 1-Hamiltonian. One may ask whethe the convese of Theoem 1 holds. To answe this question, we conside the gaph M shown in Figue 2(a) which is constucted by a sequence of 3-vetex expansions on K 3,3. To be specific, let {a i,b i 1 i 3} be the vetex set of K 3,3 and {(a i,b j ) 1 i, j 3} be the edge set of K 3,3. The constuction of M is given as follows: 1. Pefom 3-vetex expansions on vetices a 1,a 2, and a 3, that is, eplace each a i with a K 3 given by vetex set {a i,m 1 m 3}. 2. Pefom 3-vetex expansions on vetices b 1 and b 3, that is, eplace each b i fo i =1, 3 with a K 3 given by vetex set {b i,m 1 m 3}. Since K 3,3 and K 4 ae 1-edge-Hamiltonian, it follows fom Coollay 1 that M is also 1-edge-Hamiltonian. Note that M b 2 is not Hamiltonian. It follows that M is not 1-Hamiltonian. Futhemoe, it can be easily veified that V (M) ={b 2 }. By pefoming a 3-vetex expansion on b 2 of M, we obtain a new gaph M as shown in Fig- FIG. 2. A counteexample fo the convese of Theoem 1. ue 2(b). In othe wods, M is a 3-join of M and K 4. Note that M is 1-Hamiltonian following fom Coollay 2, wheeas M is not. This povides a counteexample fo the convese of Theoem 1. Moeove, note that a 3-join opeation on a 1-Hamiltonian gaph and a non-1- Hamiltonian gaph may geneate a 1-Hamiltonian gaph Families of 1-Hamiltonian 3-Join Gaphs In this subsection, we show that known tivalent 1- Hamiltonian gaphs poposed in [3 5] and those with even vetices poposed in [1,2] can be geneated by a sequence of 3-joins on some specific gaphs and K 4. Haay and Hayes [1,2] poposed a family of 1- Hamiltonian gaphs, denoted by H(k) fo k 4, whee V(H(k)) = {0, 1, 2,..., k 1}, and {(i, i +1) 0 i k 2} {(0,k/2), (0,k 1)} E(H(k)) = {(i, k i) 1 i k/2 1} fo k even, {(0, 1), (0, 2), (0,k 1), (0,k 2)} {(i, i +2) 1 i k 3} {(i, i +1) 1 i k 2 and i odd} fo k odd. Examples of H(4),H(5),H(8), and H(9) ae shown in Figue 3. Note that H(4) is, indeed, a complete gaph K 4 which is the smallest 1-Hamiltonian gaph. Futhemoe, H(k) fo all k 4 and even ae tivalent. The following theoem can be easily veified: Theoem 2. Let k 6 be an even intege. Given that H(4) = K 4,H(k) can be obtained by a 3-vetex expansion at the vetex 0[o (k 2)/2] in H(k 2) and then elabeling vetices. Futhemoe, H(k) is 1-Hamiltonian. NETWORKS

4 FIG. 3. The gaphs H(k). H(k) fo k odd will be discussed in Section 2.3. Mukhopadhyaya and Sinha [4] poposed a family of 1-Hamiltonian gaphs, denoted by M(k). Let k 4bean intege and t be a nonnegative intege. To define M(k), we fist intoduce B(i, t), as illustated in Figue 4, which is defined as follows: V(B(i, t)) = {x i,j 1 j t}{x l i,j 1 j t}{y i,z i }, and E(B(i, t)) = {(x l i,j,x l i,j 1) 1 <j t} {(x i,j,x i,j+1) 1 j t 1} {(xi,j,x l i,j) 1 j t} {(xi,1,y l i ), (y i,z i ), (y i,xi,1)}. It can be easily veified that B(i, t), fo t 2, is isomophic to the gaph obtained fom pefoming a 3- vetex expansion on the vetex y i of B(i, t 1). The gaph M(k) fo k even is constucted as follows. (Fo convenience, when t = 0, we wite y i = x l i,0 = x i,0.) 1. If k =6t + 4, then M(k) is constucted fom thee B(i, t) fo all 0 i 2 by identifying z 1,z 2, and z 3 into a single vetex z and adding the edges (x0,t,x l 1,t), (x1,t,x l 2,t), and (x2,t,x l 0,t). 2. If k = 6t + 6, then M(k) is constucted fom B(0,t +1),B(1,t), and B(2,t) by identifying z 1,z 2, and z 3 into a single vetex z and adding the edges (x0,t+1,x l 1,t), (x1,t,x l 2,t), and (x2,t,x l 0,t+1). 3. If k = 6t + 8, then M(k) is constucted fom B(0,t+1),B(1,t+ 1), and B(2,t) by identifying z 1,z 2, and z 3 into a single vetex z and adding the edges (x0,t+1,x l 1,t+1), (x1,t+1,x l 2,t), and (x2,t,x l 0,t+1). Note that M(4) is, indeed, a complete gaph K 4. The gaph M(k) fo k 5 and odd is constucted as follows: 1. If k =8t +5,M(k) is constucted fom fou B(i, t) fo all 0 i 3 by identifying all z i into a single vetex z and adding the edges (x0,t,x l 1,t), (x1,t,x l 2,t), (x2,t,x l 3,t), and (x3,t,x l 0,t). 2. If k = 8t +7,M(k) is constucted fom B(0,t + 1) and B(i, t) fo i = 1, 2, 3 by identifying all z i into a single vetex z and adding the edges (x0,t+1,x l 1,t), (x1,t,x l 2,t), (x2,t,x l 3,t), and (x3,t,x l 0,t+1). 3. If k = 8t +9,M(k) is constucted fom B(i, t +1) fo i =0, 1 and B(j, t) fo j =2, 3 by identifying all z i into a single vetex z and adding the edges (x0,t+1,x l 1,t+1), (x1,t+1,x l 2,t), (x2,t,x l 3,t), and (x3,t,x l 0,t+1). 4. If k = 8t + 11,M(k) is constucted fom B(i, t + 1) fo i = 0, 1, 2 and B(3, t) by identifying all z i into a single vetex z and adding the edges (x0,t+1,x l 1,t+1), (x1,t+1,x l 2,t+1), (x2,t+1,x l 3,t), and (x3,t,x l 0,t+1). The gaphs M(4),M(5),M(18), and M(25) ae shown in Figue 5. The following theoem can be easily veified: Theoem 3. Let k and t be nonnegative integes. Let M(4) and M(5) be given as in Figue 5. (i) M(k) fo k 6 and even can be obtained by a 3-vetex expansion of M(k 2) at the vetex y 0 if k =6t +6, at y 1 if k =6t +8, and at y 2 if y =6t +10. (ii) M(k) fo k 7 and odd can be obtained by a 3-vetex expansion of M(k 2) at the vetex y 0 if k =8t +7, at y 1 if k =8t +9, at y 2 if k =8t +11, and at y 3 if y =8t +13. (iii) M(k) is 1-Hamiltonian fo all k 4. Wang et al. [5] pesented a new family of 1- Hamiltonian gaphs W(k), as illustated in Figue 6, which is constucted fom B(i, k) fo 0 i 2k by joining all z i with a cycle z 0,z 1,...,z 2k,z 0 and adding edges (x l 2k,k,x 0,k) and (x l i,k,x i+1,k) fo all 0 i 2k 1. Let D(k) denote the gaph obtained fom B(i, 0) fo 0 i 2k by joining all z i and y i with cycles z 0,z 1,...,z 2k,z 0 and y 0,y 1,...,y 2k,y 0, espectively. It can be easily veified that D(k) is 1-Hamiltonian. The following theoem can also be easily veified: Theoem 4. Let k 1 be an intege. (i) W(k) can be geneated by pefoming k 3-vetex expansions at y i in D(k) fo 0 i 2k. (ii) W(k) is 1-Hamiltonian. Hung et al. [3] poposed a family of 1-Hamiltonian gaphs, called a Chistmas tee, denoted by CT(k). In thei constuction, they, indeed, employed 3-vetex ex- FIG. 4. The gaphs B(i, t). FIG. 5. The gaphs M(k). 236 NETWORKS 2000

5 FIG. 6. The gaph W(2). pansions. Let be an abitay vetex in K 4. The gaphs CT(k) ae defined as follows: CT(1) = K 4 ; fo k 2,CT(k) is ecusively constucted by pefoming a 3- vetex expansion in CT(k 1) on evey vetex which is at the distance of k 1 fom the vetex. We illustate an example of CT(3) in Figue 7. By this constuction, CT(k) is obviously 1-Hamiltonian (3,4)-Join To constuct the gaphs H(k) poposed in [1,2] fo k odd, we intoduce a vaiation of 3-join, called (3,4)-join, which is pefomed at a vetex of degee 3 in one gaph and a vetex of degee 4 in anothe gaph. Definition 3. Let x be a vetex of degee 3 in G 1 and y be a vetex of degee 4 in G 2. Let N(x) ={x 1,x 2,x 3 } and N(y) ={y 1,y 2,y 3,y 4 }. The (3,4)-join of G 1 and G 2 is defined as the gaph K such that V(K) = (V(G 1 ) {x}) (V(G 2 ) {y}) E(K) = (E(G 1 ) {(x, x i ) 1 i 3}) (E(G 2 ) {(y, y i ) 1 i 4}) {(x i,y i ) 1 i 3}{(x 3,y 4 )}. and We call a gaph K a (3,4)-join of G 1 and G 2 if K is a (3,4)-join at some vetex x of degee 3 in G 1 and some vetex y of degee 4 in G 2 with some specific N(x) and N(y). Simila to the 3-join opeation, each (3,4)-join of G 1 and G 2 is uniquely defined by N(x) and N(y). Since we ae inteested in 1-Hamiltonian gaphs, we estict ou discussion on (3,4)-join to paticula gaphs, whee (i) G 1 and G 2 ae 1-Hamiltonian, (ii) G 1 has one vetex x of degee 3, and (iii) G 2 has one vetex y of degee 4. Note that not all N(x) and N(y) can induce 1- Hamiltonian gaphs by using (3,4)-join on G 1 and G 2. Fo example, Figue 8(a) and (b) illustates two (3,4)- joins of G 1 = K 4 and G 2 = H(5) with diffeent N(x) and N(y). Given N(x) ={x 1,x 2,x 3 } in G 1,N(y) ofg 2 in Figue 8(a) and (b) ae given by {y 1,y 2,y 3,y 4 } and {y 1,y 3,y 2,y 4 }, espectively. It can be obseved that the gaph in Figue 8(a) is 1-Hamiltonian, but the one in Figue 8(b) is not 1-Hamiltonian since it is not Hamiltonian when deleting the vetex x 3 fom the gaph. The odeing of N(x) and N(y) plays an impotant ole in detemining the 1-Hamiltonicity of the esulting (3,4)-join. It is inteesting to find sufficient conditions of N(x),N(y),G 1, and G 2 fo the esulting (3,4)-join to have a Hamiltonian popety. Fo example, let N(x) = {x 1,x 2,x 3 } and N(y) ={y 1,y 2,y 3,y 4 }.IfG 1 and G 2 have a Hamiltonian cycle containing the edges (x, x i ), (x, x j ) and (y, y i ), (y, y j ), espectively, with i, j {1, 2, 3} and i j, then the esulting (3,4)-join is also Hamiltonian. In the following theoem, we give a sufficient condition of N(x),N(y),G 1, and G 2 fo the esulting (3,4)-join to be 1-Hamiltonian. Theoem 5. Let G 1 and G 2 be two 1-Hamiltonian gaphs, whee G 1 has a vetex x of degee 3 with N(x) ={x 1,x 2,x 3 } and G 2 has a vetex y of degee 4 with N(y) ={y 1,y 2,y 3,y 4 }. Let f 1 V(G 1 ) E(G 1 ) and f 2 V(G 2 ) E(G 2 ) with f 1 x and f 2 y. Suppose that G 1,G 2,N(x), and N(y) satisfy the following conditions: FIG. 7. The Chistmas tee CT(3). FIG. 8. Examples of (3,4)-joins of gaphs K 4 and H(5), whee the gaph in (a) is 1-Hamiltonian and the one in (b) is not. NETWORKS

6 (A1) Fo evey Hamiltonian cycle in G 1 f 1 without containing an edge (x, x i ) with i {1, 2}, thee is a Hamiltonian cycle in G 2 (y, y i ) containing the edge (y, y 3 i ); (A2) Fo evey Hamiltonian cycle in G 1 f 1 without containing the edge (x, x 3 ), thee is a Hamiltonian cycle in G 2 containing the edges (y, y 1 ) and (y, y 2 ); (A3) Fo evey Hamiltonian cycle in G 2 f 2 containing the edges (y, y 3 ) and (y, y 4 ) with f 2 / {y 1,y 2 }, thee is a Hamiltonian path in G 2 f 2 fom y 1 to y 2 containing (y, y 3 ) and (y, y 4 ); (A4) Thee is a Hamiltonian cycle of G 2 y i containing the edge (y, y 3 i ) fo i {1, 2}. Let K be the (3,4)-join of G 1 and G 2 at x and y with the given N(x) and N(y). Then, K is a 1-Hamiltonian gaph. Poof. Let f be any fault, vetex o edge of K. We distinguish the following cases of f: Case 1. f (V(G 1 ) {x})(e(g 1 ) {(x, x i ) 1 i 3}). Since G 1 is 1-Hamiltonian and x is of degee 3, thee is a Hamiltonian cycle H 1 in G 1 f without containing an edge (x, x i ), whee i {1, 2, 3}. Conside i {1, 2}. Then, H 1 can be witten as H 1 = x, x 3 P 1 x 3 i,x, whee P 1 is a path in G 1. By (A1), thee is a Hamiltonian cycle H 2 in G 2 (y, y i ) containing the edge (y, y 3 i ) which can be witten as H 2 = y, y 3 i P 2 y j,y, whee j {3, 4} and P 2 is a path in G 2. Then, x 3 i,y 3 i P 2 y j,x 3 P 1 x 3 i is a Hamiltonian of K f. Conside i = 3. Then, H 1 can be witten as H 1 = x, x 2 P 1 x 1,x, whee P 1 is a path in G 1. By (A2), thee is a Hamiltonian cycle H 2 in G 2 containing the edges (y, y 1 ) and (y, y 2 ) which can be witten as H 2 = y, y 1 P 2 y 2,y, whee P 2 is a path in G 2. Thus, x 1,y 1 P 2 y 2,x 2 P 1 x 1 is a Hamiltonian cycle of K f. Case 2. f (V(G 2 ) {y})(e(g 2 ) {(y, y i ) 1 i 4}). Conside f/ {y 1,y 2 }. Let H 2 be a Hamiltonian cycle in G 2 f. We fist assume that H 2 contains both (y, y 3 ) and (y, y 4 ). By (A3), thee is a Hamiltonian path P 2 in G 2 f fom y 1 to y 2 containing (y, y 3 ) and (y, y 4 ). Since G 1 is 1-Hamiltonian, thee is a Hamiltonian cycle H 1 of G 1 x 3 which can be witten as x, x 2 P 1 x 1,x, whee P 1 is a path of G 1 x 3. Let P 3 denote the path obtained fom P 2 by eplacing the edges (y, y 3 ) and (y, y 4 ) with (x 3,y 3 ) and (x 3,y 4 ). Then, y 1 P 3 y 2,x 2 P 1 x 1,y 1 is a Hamiltonian cycle of K f. Assume that H 2 contains at most one of (y, y 3 ) and (y, y 4 ). Since y is of degee 4, H 2 contains at least one of (y, y 1 ) and (y, y 2 ). If H 2 contains both (y, y 1 ) and (y, y 2 ), then let H 1 be a Hamiltonian cycle of G 1 (x, x 3 ). If H 2 contains (y, y i ) fo i = 1 o 2, then let H 1 be a Hamiltonian cycle of G 1 (x, x 3 i ). Using simila aguments in Case 1, we can find a Hamiltonian cycle in K f. Conside f {y 1,y 2 }. By (A4), thee is a Hamiltonian cycle in G 2 y i containing the edge (y, y 3 i ), whee i {1, 2}. In othe wods, this Hamiltonian cycle contains eithe (y, y 3 )o(y, y 4 ). Simila to the above-mentioned case, we can find a Hamiltonian cycle in K f. Case 3. f {(x i,y i ) 1 i 3}{(x 3,y 4 )}. Conside f =(x 1,y 1 )o(x 2,y 2 ). Let H 1 be a Hamiltonian cycle of G 1 (x, x i ) with i {1, 2} which can be witten as H 1 = x, x 3 i P 1 x 3,x, whee P 1 is a path in G 1. By (A1), thee is a Hamiltonian cycle H 2 of G 2 (y, y i ) containing the edge (y, y 3 i ) which is given by H 2 = y, y j P 2 y 3 i,y, whee j {3, 4} and P 2 is a path in G 2. Then, x 3,y j P 2 y 3 i,x 3 i P 1 x 3 is a Hamiltonian cycle of K f. Conside f =(x 3,y 3 )o(x 3,y 4 ). Let H 1 be a Hamiltonian cycle of G 1 (x, x 3 ) which can be witten as H 1 = x, x 2 P 1 x 1,x, whee P 1 is a path in G 1. By (A2), thee is a Hamiltonian cycle H 2 in G 2 containing the edges (y, y 1 ) and (y, y 2 ) which is given by H 2 = y, y 1 P 2 y 2,y, whee P 2 is a path in G 2. Then, x 1,y 1 P 2 y 2,x 2 P 1 x 1 is a Hamiltonian cycle of K f. Hence, this theoem follows. Howeve, it is, in geneal, difficult to veify whethe G 1 and G 2 satisfy the conditions (A1), (A2), (A3), and (A4). Nonetheless, some specific gaphs can be easily veified to satisfy these conditions. An example is shown in the poof of the following theoem which states the constuction of H(k) fo k 5 and odd as poposed in [1,2]: Theoem 6. Let {x, x 1,x 2,x 3 } be the vetex set of K 4. Let H(5) be given as shown in Figue 3. Then, H(k) fo k 7 and odd can be obtained by a (3,4)-join at the vetex x in K 4 and the vetex 0 in H(k 2). Futhemoe, H(k) is 1-Hamiltonian. Poof. Although the odeing of N(x) ink 4 can be abitay, we abitaily define N(x) ={x 1,x 2,x 3 }. Let N(0) in H(5) be given by N(0) = {1, 2, 3, 4}. By pefoming a (3,4)-join on K 4 and H(5) at the vetices x and 0, we obtain a gaph which is isomophic to H(7) by elabeling x 3 with 0, x 1 with 1, x 2 with 2, and i with i +2 fo 1 i 4. Suppose that H(l) is a (3,4)-join on K 4 and H(l 2) fo l 7. Let N(0) in H(l) begivenbyn(0) = {1, 2, l 2, l 1}. Similaly, by pefoming a (3,4)-join on the vetex x in K 4 and the vetex 0 in H(l), we obtain a gaph which is isomophic to H(l + 2) by elabeling x 3 with 0, x 1 with 1, x 2 with 2, and i with i + 2 fo all 1 i l 1. Hence, H(k) can be obtained fom K 4 and H(k 238 NETWORKS 2000

7 2) by pefoming a (3,4)-join on the vetex x and the vetex 0. It can be veified that H(5) is 1-Hamiltonian. To pove that H(k) fo k 7 is 1-Hamiltonian, we need to show that (A1), (A2), (A3), and (A4) ae satisfied. To this end, let f 1 {x 1,x 2,x 3 }E(K 4 ),f 2 (V(H(k)) {0}) E(H(k)), and N(0) = {1, 2,k 2,k 1} in H(k). To show that conditions (A1) and (A2) ae satisfied, we need to constuct a Hamiltonian cycle in H(k) (0, 1) containing the edge (0, 2), a Hamiltonian cycle in H(k) (0, 2) containing the edge (0, 1), and a Hamiltonian cycle in H(k) containing the two edges (0, 1) and (0, 2). The desied Hamiltonian cycle in H(k) (0, 1) can be constucted as follows: 1, 3, 4, 6, 5, 7, 8,...,k 3,k 1,k 2, 0, 2, 1 if (k 1)/2 is odd, 1, 3, 4, 6, 5, 7, 8,...,k 4,k 2,k 1, 0, 2, 1 if (k 1)/2 iseven. Also, the desied Hamiltonian cycle in H(k) (0, 2) can be constucted as follows: 1, 2, 4, 3, 5, 6,...,k 4,k 2,k 1, 0, 1 if (k 1)/2 is odd, 1, 2, 4, 3, 5, 6,...,k 3,k 1,k 2, 0, 1 if (k 1)/2 iseven. Futhemoe, we can constuct a Hamiltonian cycle in H(k) as follows: 0, 1, 3,...,k 2,k 1,k 3,...,2, 0, which contains the two edges (0, 1) and (0, 2). Theefoe, (A1) and (A2) ae satisfied. Let H be any Hamiltonian cycle in H(k) f 2 fo f 2 (V(H(k)) {0, 1, 2})E(H(k)) containing the edges (0,k 2) and (0,k 1). Since the vetices 1 and 2 ae of degee 3, H contains 3, 1, 2, 4 as a subpath. Then, H (1, 2) is the desied Hamiltonian path in H(k) f 2 containing the edges (0,k 2) and (0,k 1). It follows that (A3) is satisfied. To satisfy (A4), we constuct a Hamiltonian cycle of H(k) 1 containing (0, 2) as follows: 3, 5, 6, 8, 7,...,k 2,k 1, 0, 2, 4, 3 if (k 1)/2 is odd, 3, 5, 6, 8, 7,...,k 3,k 1,k 2, 0, 2, 4, 3 if (k 1)/2 iseven, Theefoe, H(k) is 1-Hamiltonian following fom ecusively applying Theoem 5. Theoem 6 states that H(k) fo k 7 and odd poposed by Haay and Hayes [1,2] can be obtained fom a (3,4)-join of K 4 and H(k 2). 3. CYCLE EXTENSION Let G be a gaph and C = x 0,x 1,...,x k 1,x 0 be a cycle of G, whee k 3 is an abitay intege. We intoduce an opeation called cycle extension which includes two aspects: Fist, augment G by eplacing each edge in C with a path of odd length and, second, add a new cycle to the augmented gaph in a specific manne. To be specific, we define cycle extension as follows: Definition 4. The cycle extension of G aound C is a gaph denoted by Ext C (G) and given as follows: V(Ext C (G)) = {p i,j,q i,j 1 j l i }V(G), and E(Ext C (G)) = (E(G) E(C)) {(p i,j,q i,j ) 1 j l i } [{(x i,p i,1 ), (p i,li,x i+1 )} {(p i,j,p i,j+1 ) 1 j l i 1}] [{(q i,j,q i,j+1 ) 1 j l i 1} {(q i,li,q i+1,1 )}], whee l i is even fo all i. We call the cycle induced by the vetices {q i,j 1 j l i } the extended cycle of C, denoted by O C. An example of Ext C (G) is illustated in Figue 9, whee C is epesented by dakened edges, and O C, by dashed edges. Thoughout this section, we adopt the following notion: and the desied Hamiltonian cycle of H(k) 2 containing (0, 1) is 4, 6, 5, 7, 8,...,k 3,k 1,k 2, 0, 1, 3, 4 if (k 1)/2 is odd, 4, 6, 5, 7, 8,...,k 4,k 2,k 1, 0, 1, 3, 4 if (k 1)/2 iseven. It follows that (A4) is satisfied. FIG. 9. An example of Ext C (G). NETWORKS

8 C = x 0,x 1,...,x k 1,x 0, l i : an even intege which is the numbe of vetices in Ext C (G) added between x i and x i+1, p i,j : a vetex in Ext C (G) added between x i and x i+1, and q i,j : the vetex in Ext C (G) that is adjacent to p i,j. Many known families of 1-Hamiltonian gaphs ae tivalent gaphs. It can be easily veified that if G is a tivalent gaph then Ext C (G) is also a tivalent gaph. Since we ae inteested in constucting 1-Hamiltonian gaphs, we focus ou discussion of cycle extensions on tivalent gaphs only. Hencefoth, we suppose that G is tivalent in the following discussion of cycle extensions. In the following discussion, we use z i to denote the unique neighbo of x i that is not in C. The addition and subtaction involved in the subscipt o index of a vetex in a cycle is taken modulo k, whee k denotes the length of the cycle o k is clea fom context without ambiguity. Fo example, (x i,x i+1 ) fo i = k 1inC is simply (x k 1,x 0 ), and (q i 1,li 1,q i,1 )ino C fo i = 0 is simply (q k 1,lk 1,q 0,1 ). Let H be a cycle of G. WeuseH C to denote the cycle in Ext C (G) obtained fom H by eplacing all (x j, x j+1 ) in E(H) E(C) with x j, p j,1, p j,2,...,p j,lj, x j+1.we use Ω H to denote the cycle obtained fom O C by eplacing evey q i,1, q i,2, q i,3,...,q i,li with q i,1, p i,1, p i,2, q i,2, q i,3, p i,3,...,p i,li, q i,li if p i,1,p i,2,...,p i,li is not a subpath in H C. In what follows, we intoduce six opeations M 1 (H, e, j),m 2 (H, e), and M i (H, x) fo 3 i 6 that augment the cycle H of G to a cycle of Ext C (G) with espect to some edge e =(x i,x i+1 ) o vetex x = x i o x = x i+1 in C and a specific j. WeuseM 1 (H, e, j),m 2 (H, e), and M i (H, x) to mean the opeation and the coesponding cycle intechangeably. Fo ease of exposition, we define Ω qi = q i,1,p i,1,p i,2,q i,2,q i,3,p i,3,...,p i,li,q i,li and Ω pi = p i,1,q i,1,q i,2,p i,2,p i,3,q i,3,q i,li,p i,li, which will be used in the definitions of opeations M 2,M 3,M 4,M 5, and M 6. To be specific, the six opeations ae defined as follows: Opeation M 1. Given a cycle H of G which contains the edge e, we define an opeation M 1 (H, e, j) to con- FIG. 11. Illustation fo opeation M 2. stuct a cycle in Ext C (G) {(p i,j,p i,j+1 ), (q i,j,q i,j+1 )} fo some 1 j l i 1. Let Q be the path Ω H (q i,j,q i,j+1 ) and P be the path H C (p i,j,p i,j+1 ). We define M 1 (H, e, j) = p i,j P p i,j+1,q i,j+1 Q q i,j,p i,j, as illustated in Figue 10. Opeation M 2. Given that z i and z i+1 ae adjacent in G and given a Hamiltonian cycle H of G containing x i 1,x i,x i+1,x i+2 as a subpath, we define an opeation M 2 (H, e) to constuct a Hamiltonian cycle of Ext C (G) {(q i 1,li 1,q i,1 ), (q i,li,q i+1,1 )}. Let y i be the unique neighbo of z i diffeent fom x i and z i+1, and y i+1 be the unique neighbo of z i+1 diffeent fom x i+1 and z i. Since G is tivalent and H contains x i 1,x i,x i+1,x i+2 as a subpath, y i,z i,z i+1,y i+1 is a subpath of H in G. Moeove, Ω H contains q i 1,li 1,q i,1,...,q i,li,q i+1,1 as a subpath and HC contains p i 1,li 1,x i,p i,1,p i,2,...,p i,li,x i+1,p i+1,1 as a subpath. Hence, we can obtain a path P fom HC by eplacing p i 1,li 1,x i,p i,1,p i,2,...,p i,li,x i+1,p i+1,1 with x i,p i,1 Ω pi p i,li,x i+1 and eplacing (z i,z i+1 ) with (z i,x i ) and (z i+1,x i+1 ). Deleting q i 1,li 1,q i,1,...,q i,li,q i+1,1 fom Ω H yields a path Q. Then, we define M 2 (H, e) = p i 1,li 1,q i 1,li 1 Q q i+1,1,p i+1,1 P p i 1,li 1, as illustated in Figue 11. FIG. 10. Illustation fo opeation M 1. FIG. 12. Illustation fo opeations M 3 and M NETWORKS 2000

9 Opeations M 3 and M 4. Given a Hamiltonian cycle H of G x i+1, we define an opeation M 3 (H, x i+1 ) to constuct a Hamiltonian cycle of Ext C (G) p i,j with j odd and an opeation M 4 (H, x i+1 ) to constuct a Hamiltonian cycle of Ext C (G) q i,j with j even. Since x i+1 is not in H, H contains x i 1,x i,z i and x i+3,x i+2,z i+2 as subpaths. See Figue 12(a) fo an illustation. Moeove, q i,1 Ω qi q i,li,q i+1,1 Ω qi+1 q i+1,li+1 is a subpath of Ω H. Let Q =Ω H q i,1 Ω qi q i,li,q i+1,1 Ω qi+1 q i+1,li+1,q i+2,1, which is a path fom q i+2,1 to q i,1. Let P = HC (x i+2,p i+2,1 ), which is a path fom x i+2 to p i+2,1. Then, we define M 3 (H, e) = p i+2,1,q i+2,1 Q q i,1,p i,1,p i,2,q i,2,..., p i,j 1,q i,j 1,q i,j,q i,j+1,p i,j+1,p i,j+2, q i,j+2,...,q i,li,p i,li,x i+1,p i+1,1 Ω pi+1 p i+1,li+1,x i+2 P p i+2,1, M 4 (H, e) = p i+2,1,q i+2,1 Q q i,1,p i,1,p i,2,q i,2,..., q i,j 1,p i,j 1,p i,j,p i,j+1,q i,j+1,q i,j+2, p i,j+2,...,p i,li,x i+1,p i+1,1 Ω pi+1 p i+1,li+1,x i+2 P p i+2,1, as illustated in Figue 12(b) and (c). Opeations M 5 and M 6. Given a Hamiltonian cycle H of G x i, as illustated in Figue 13(a), we define an opeation M 5 (H, x i ) to constuct a Hamiltonian cycle of Ext C (G) p i,j with j even and an opeation M 6 (H, x i )to constuct a Hamiltonian cycle of Ext C (G) q i,j with j odd. Let Q = Ω H q i 2,li 2, q i 1,1 Ω qi 1 q i 1,li 1, q i,1 Ω qi q i,li, which is a path fom q i,li to q i 2,li 2. Let P = HC (x i 1,p i 2,li 2 ), which is a path fom p i 2,li 2 to x i 1. We define M 5 (H, x) = x i 1,p i 1,1 Ω pi 1 p i 1,li 1,x i,p i,1,q i,1, q i,2,p i,2,p i,3,...,p i,j 1,q i,j 1,q i,j,q i,j+1, p i,j+1,p i,j+2,q i,j+2,...,q i,li Q q i 2,li 2, p i 2,li 2 P x i 1, M 6 (H, x) = x i 1,p i 1,1 Ω pi 1 p i 1,li 1,x i,p i,1, q i,1,q i,2,p i,2,p i,3,...,q i,j 1,p i,j 1,p i,j,p i,j+1, q i,j+1,q i,j+2,p i,j+2,...,q i,li Q q i 2,li 2, p i 2,li 2 P x i 1, H of G f such that at least an edge in C, say (x i,x i+1 ), is in H. Theefoe, M 1 (H, (x i,x i+1 ),j), 1 j l i 1, foms a Hamiltonian cycle of Ext C (G) f. Conside f = x i fo 0 i k 1. Since G is tivalent and k 3, two vetices x i 1 and x i+1 in C have degee 2 in G f. It follows that we can always find a Hamiltonian cycle H of G f such that H contains an edge in C, say (x j,x j+1 ), fo j i 1,i. On the othe hand, Ω H contains the subpath q i 2,li 2,q i 1,1 Ω qi 1 q i 1,li 1,q i,1 Ω qi q i,li,q i+1,1 which does not contain the vetex x i. Theefoe, M 1 (H, (x j,x j+1 ),j ), 1 j l j 1, is a Hamiltonian cycle of Ext C (G) f. Conside f {p i,j,p i,j,q i,j,q i,j } fo some 1 j, j l i with j odd and j even. Since G is 1-vetex- Hamiltonian, thee ae Hamiltonian cycles H 1 and H 2 of G x i+1 and G x i, espectively. It follows that the opeations M 3 (H 1,x i+1 ),M 4 (H 1,x i+1 ),M 5 (H 2,x i ), and M 6 (H 2,x i ) can be applied and they ae, indeed, Hamiltonian cycles of Ext C (G) p i,j,ext C (G) q i,j,ext C (G) p i,j, and Ext C (G) q i,j, espectively. Hence, the lemma follows. Lemma 2. Let G be a tivalent 1-edge-Hamiltonian gaph and C = x 0,x 1,...,x k 1,x 0 be a cycle of G. If f E(Ext C (G)) {(q i,li,q i+1,1 )}, then Ext C (G) f is Hamiltonian. Poof. Conside that f is an edge in E(G) E(C). Since G is tivalent 1-edge-Hamiltonian, thee is a Hamiltonian cycle H of G f such that H contains at least an edge in C, say (x i,x i+1 ), with 0 i k 1. It follows that M 1 (H, (x i,x i+1 ),j) with 1 j l i 1 can be applied and yields a Hamiltonian cycle of Ext C (G) f. Conside f = (x i,p i,1 )o(p i,li,x i+1 ) fo some 0 i k 1. Since G is 1-edge-Hamiltonian, thee is a Hamiltonian cycle H of G (x i,x i+1 ). It follows fom the tivalence of G that H contains (x i 1,x i ) and (x i+1,x i+2 ). Futhemoe, Ω H contains q i 1,li 1,q i,1 Ω qi q i,li,q i+1,1 as a subpath. On the othe hand, both subpaths x i 1,x i,z i and z i+1,x i+1,x i+2 ae in H. Then, we can apply the opeation M 1 on (x i 1,x i ) and j fo as illustated in Figue 13(b) and (c). These six opeations ae used in the poofs of the following lemmas and theoems: Lemma 1. Let G be a tivalent 1-vetex-Hamiltonian gaph and C = x 0,x 1,...,x k 1,x 0 be a cycle of G. Then, Ext C (G) is tivalent 1-vetex-Hamiltonian. Poof. Conside f V(G) V(C). Since G is tivalent 1-vetex-Hamiltonian, thee is a Hamiltonian cycle FIG. 13. Illustation fo Opeations M 5 and M 6. NETWORKS

10 1 j l i 1 1. It follows that M 1 (H, (x i 1,x i ),j) foms a Hamiltonian cycle of Ext C (G) {(x i,p i,1 ), (p i,li,x i+1 )}. Conside f = (p i,j,p i,j+1 ) o (q i,j,q i,j+1 ) fo some 0 i k 1 and some 1 j l i 1. Let H be a Hamiltonian cycle of G (x i 1,x i ). It follows that (x i,x i+1 )isinh. Theefoe, M 1 (H, (x i,x i+1 ),j) with 1 j l i 1 foms a Hamiltonian cycle of Ext C (G) {(p i,j,p i,j+1 ), (q i,j,q i,j+1 )}. Conside f = (p i,j,q i,j ) fo some 0 i k 1 and some 1 j l i. Let H be a Hamiltonian cycle of G (x i 1,x i ). It follows that both (x i,x i+1 ) and (x i 2,x i 1 ) ae in H. Moeove, H C contains x i,p i,1,p i,2,...,p i,li,x i+1 as a subpath and Ω H contains q i 1,li 1,q i,1,q i,2,...,q i,li,q i+1,1 as a subpath. Hence, M 1 (H, (x i 2,x i 1 ),j ) fo 1 j l i 2 1 foms a Hamiltonian cycle of Ext C (G) (p i,j,q i,j ). The lemma is poved. One may ask whethe Ext C (G) is 1-Hamiltonian if G is tivalent 1-Hamiltonian. The answe is no and it can be veified by a counteexample shown in Figue 14. The gaphs shown in Figue 14, poposed by Wang et al. [6], ae called eye netwoks and denoted by Eye(n) fo n 1. The gaph Eye(1) shown in Figue 14 is a tivalent 1-Hamiltonian gaph. Let O 1 be the cycle indicated by daken edges in Eye(1) as shown in Figue 14. Note that Eye(2) = Ext O1 (Eye(1)). Although Eye(1) is tivalent 1-Hamiltonian, Eye(2) is not 1-Hamiltonian since we cannot find a Hamiltonian cycle in Eye(2) ((2, 0), (2, 3)),Eye(2) ((2, 6), (2, 9)), and Eye(2) ((2, 12), (2, 15)). Although that G is tivalent 1-Hamiltonian does not imply that Ext C (G) is 1-Hamiltonian, we ae inteested in finding a sufficient condition on cycle C fo Ext C (G)to be 1-Hamiltonian. To this end, we define the ecoveable set R(C) of cycle C as follows: R(C) = {(x i,x i+1 ) x i 1,x i,x i+1,x i+2 is a subpath of some Hamiltonian cycle of G and (z i,z i+1 ) is an edge in G fo 0 i k 1}. FIG. 15. The cycle x 0,x 1,x 2,x 3,x 0 is ecoveable with espect to the gaph G. The definition of R(C), indeed, aises fom the given condition fo the opeation M 2 to be applicable. A cycle C of G is said to be ecoveable with espect to G if no two edges of E(C) R(C) ae adjacent. Fo example, the cycle C = x 0,x 1,x 2,x 3,x 0 shown in Figue 15 is ecoveable with espect to G since {(x 0,x 1 ), (x 2,x 3 )} is the ecoveable set of C. Theoem 7. Let G be a tivalent 1-Hamiltonian gaph and C = x 0,x 1,...,x k 1,x 0 be ecoveable with espect to G. Then, Ext C (G) is tivalent 1-Hamiltonian. Poof. It follows fom Lemma 1 that Ext C (G) is a 1-vetex-Hamiltonian gaph. Futhemoe, it follows fom Lemma 2 that Ext C (G) f is Hamiltonian fo f E(Ext C (G)) {(q i,li,q i+1,1 )}. Now, it suffices to show that Ext C (G) f is Hamiltonian fo f {(q i,li,q i+1,1 )}. Equivalently, we will constuct a Hamiltonian cycle in Ext C (G) without using (q i,li,q i+1,1 ) fo 0 i k 1. Since C is ecoveable with espect to G, it follows that R(C) and we have eithe (x i,x i+1 ) R(C) o (x i,x i+1 ) / R(C). Fo (x j,x j+1 ) R(C), we use H j to denote a Hamiltonian cycle of G such that H j contains x j 1,x j,x j+1,x j+2 as a subpath. Fist, conside (x i,x i+1 ) R(C). Then, the opeation M 2 can be applied and M 2 (H i, (x i,x i+1 )) is a Hamiltonian cycle of Ext C (G) {(q i 1,li 1,q i,1 ), (q i,li,q i+1,1 )}. Next, conside (x i,x i+1 ) / R(C). Since any two edges in E(C) R(C) ae not adjacent, it follows that (x i+1,x i+2 ) is in R(C). Then, M 2 (H i+1, (x i+1,x i+2 )) foms a Hamiltonian cycle of Ext C (G) {(q i,li,q i+1,1 ), (q i+1,li+1,q i+2,1 )} fo (x i+1,x i+2 ) R(C). Theefoe, when G is 1-Hamiltonian and C is ecoveable with espect to G, Ext C (G) is1- edge-hamiltonian. Thus, the theoem follows. FIG. 14. Eye netwoks. Although it is, in geneal, had to veify whethe a cycle C is ecoveable with espect to G, we show that O C is ecoveable with espect to Ext C (G) in the following lemma: Lemma 3. Let G be a tivalent gaph and C = x 0,x 1,...,x k 1,x 0 be a cycle of G. If G (x i,x i+1 ) is Hamiltonian fo evey 0 i k 1, then O C is ecoveable with espect to Ext C (G). 242 NETWORKS 2000

11 Poof. It suffices to show that R(O C ) is a collection of (q i,j,q i,j+1 ) fo all 0 i k 1 and 1 j l i 1 since each p i,j is adjacent to p i,j+1 in Ext C (G) butp i,li is not adjacent to p i+1,1. Let H be a Hamiltonian cycle of G (x i 1,x i ). Since G is tivalent, both (x i 2,x i 1 ) and (x i,x i+1 ) ae in H. It follows that Ω H contains q i 3,li 3, q i 2,1, q i 2,2,...,q i 2,li 2, q i 1,1 and q i 1,li 1, q i,1, q i,2,...,q i,li, q i+1,1 as subpaths. Consequently, the Hamiltonian cycle M 1 (H, (x i 2,x i 1 ),j ), 1 j l i 2 1, of Ext C (G) contains q i 1,li 1,q i,1,q i,2,...,q i,li,q i+1,1 as a subpath and, futhemoe, (q i,j,q i,j+1 ) satisfies the definition of ecoveable set R(O C ). Since (x i 1,x i ) is an abitay edge and (q i,li,q i+1,1 ) and (q i,l i,q i +1,1) ae not adjacent fo i i,r(o C ) is a collection of (q i,j,q i,j+1 ) fo all 0 i k 1 and 1 j l i 1 and, moeove, O C is ecoveable with espect to Ext C (G). Hence, the lemma is poved. The cycle extension opeation can be ecusively pefomed: Let ExtC(G) 0 =G, OC 0 = C, ExtC(G) 1 =Ext C (G), and OC 1 = O C. Then, ExtC(G) n fo n 2 can be ecusively defined by setting Ext n C(G) =Ext O n 1 C and O n C being the extended cycle of O n 1 C (Extn 1 C in ExtC(G). n (G)) Now, conside, in paticula, Ext 2 C(G), which is the cycle extension of Ext C (G) aound O C. Given 0 i k 1 and 1 j l i 1, let l i,j and L i denote the numbe of vetices added between q i,j and q i,j+1 and between q i,li and q i+1,1, espectively. These vetices ae denoted by m i,j, whee 1 m l i,j o 1 m L i. The vetex in O 2 C which is adjacent to m i,j is denoted by s m i,j. Note that l i,j and L i ae even. Fo ease of exposition, let L = l i,j fo 1 j l i 1 and L = L i fo j = l i. To be specific, the gaph Ext 2 C(G) is given as follows (see Figue 16): V(Ext 2 C(G)) = V(Ext C (G)) 1 j l i {i,j,s m i,j 1 m m L } and E(ExtC(G))=(E(Ext 2 C (G)) E(O C )) 1 j l i ({(i,j, m i,j m+1 ), (si,j,s m i,j m+1 ) 1 m L 1} {(i,j,s m i,j) 1 m m L } 1 j l i {(q i,li,i,l 1 i ), ( L i,q i+1,1 ), (s L i,si+1,1)} 1 1 j l i 1 {(q i,j, 1 i,j), ( l i,j i,j,q i,j+1 ), Coollay 3. Let G be a tivalent gaph and C = x 0,x 1,...,x k 1,x 0 be a cycle of G. If G (x i,x i+1 ) is Hamiltonian fo all 0 i k 1, then O 2 C is ecoveable with espect to Ext 2 C(G). Lemma 4. Let G be a tivalent 1-Hamiltonian gaph and C = x 0,x 1,x 2,...,x k 1,x 0 be a cycle of G. Then, ExtC(G) 2 is tivalent 1-Hamiltonian. Poof. It follows fom Lemma 1 that ExtC(G) 2 is1- vetex-hamiltonian. It suffices to show that ExtC(G) 2 is 1-edge-Hamiltonian. We divide the edge set of ExtC(G) 2 into fou sets: E 1 = E(G) E(C), E 2 = {(q i,li,i,l 1 i ), ( L i,q i+1,1 ), (s L i,si+1,1)} 1 E 3 = 1 j l i 1 {(s l i,j i,j,s 1 i,j+1)} {(x i,p i,1 ), (p i,li,x i+1 )} {(p i,j,p i,j+1 ), (q i,j,i,j), 1 ( l i,j i,j,q i,j+1 ) } 1 j l i 1 {(i,j, m i,j m+1 ), (si,j,s m i,j m+1 ) 1 j l i 1 m L 1}, and E 4 = {(p i,j,q i,j )} 1 j l i {(i,j,s m i,j) 1 m m L }. 1 j l i Conside f E 1. By Lemma 2, thee is a Hamiltonian cycle H in Ext C (G) f. Since Ext C (G) is tivalent, it follows that H contains at least an edge e in O C, say e = (q i,j,q i,j+1 ), fo some 0 i k 1 and 1 j l i 1. Then, M 1 (H, e, m), 1 m l i,j 1, is a Hamiltonian cycle of Ext 2 C(G) f fo f E 1. Conside f E 2. Since G is 1-Hamiltonian, thee is a Hamiltonian cycle in G (x i 1,x i ). It follows fom the poof of Lemma 3 that we have (s l i,j i,j,s 1 i,j+1)}. Using the simila poof technique in Lemma 3, we can show the following coollay: FIG. 16. An example of Ext 2 C(G). NETWORKS

12 a Hamiltonian cycle, say H, in Ext C (G) containing q i 1,li 1,q i,1,q i,2,...,q i,li 1,q i,li,q i+1,1 as a subpath. Since p i,j is adjacent to p i,j+1 fo all 1 j l i 1, we can apply the opeation M 2 on any (q i,j,q i,j+1 ). Hence, M 2 (H, (q i,j,q i,j+1 )) foms a Hamiltonian cycle in ExtC(G) 2 {(s l i,j 1 i,j 1,si,j), 1 (s l i,j i,j,si,j+1)}. 1 Obseving the Hamiltonian cycle M 2 (H, (q i,li 1,q i,li )) which contains si,l 1 i,i,l 1 i,i,l 2 i,..., L i as a subpath, it follows that it is also a Hamiltonian cycle of ExtC(G) 2 (q i,li,i,l 1 i ). Similaly, we can use a Hamiltonian cycle of G (x i,x i+1 ) to constuct a Hamiltonian cycle H of Ext C (G) which contains q i,li,q i+1,1,q i+1,2,...,q i+1,li+1,q i+2,1 as a subpath. Theefoe, M 2 (H, (q i+1,1,q i+1,2 )) foms a Hamiltonian cycle in ExtC(G) 2 {( L i,q i+1,1 ), (s L i,si+1,1)}. 1 Conside f E 3. We fist conside f = (x i,p i,1 ) o (p i,li,x i+1 ). It follows fom the poof of Lemma 2 that thee is a Hamiltonian cycle H 1 in Ext C (G) f. Since Ext C (G) is tivalent, H 1 contains at least an edge (q i,j,q i,j+1) in O C fo some 0 i k 1,i i, and 1 j l i 1. Hence, M 1 (H 1, (q i,j,q i,j+1),m) is a Hamiltonian cycle of ExtC(G) 2 f, whee 1 m l i,j 1. Second, we conside f =(p i,j,p i,j+1 ), (q i,j,i,j), 1 o ( l i,j i,j,q i,j+1 ). It follows fom the poof of Lemma 2 that thee is a Hamiltonian cycle H 2 in Ext C (G) {(p i,j,p i,j+1 ), (q i,j,q i,j+1 )}. Since Ext C (G) is tivalent, H 2 contains (q i,j+1,q i,j+2 ). Hence, M 1 (H 2, (q i,j+1,q i,j+2 ),m) is a Hamiltonian cycle of ExtC(G) 2 {(p i,j,p i,j+1 ), (q i,j,i,j), 1 ( l i,j i,j,q i,j+1 )}, whee 1 m l i,j+1 1. Finally, we conside f =(i,j, m i,j m+1 ) o (si,j,s m i,j m+1 ). Similaly, thee is a Hamiltonian cycle H 3 in Ext C (G) (q i,j 1,q i,j ). Hence, we can apply the opeation M 1 on (q i,j,q i,j+1 ) and m with 1 m l i,j 1. Then, M 1 (H 3, (q i,j,q i,j+1 ),m) is a Hamiltonian cycle of ExtC(G) 2 f. Conside f E 4. Since G is 1-Hamiltonian, thee is a Hamiltonian cycle H of G (x i 1,x i ). Then, both (x i 2,x i 1 ) and (x i,x i+1 ) ae in H. Theefoe, H 1 = M 1 (H, (x i 2,x i 1 ),j) fo some 1 j l i 2 1 is a Hamiltonian cycle of Ext C (G), which contains q i 1,li 1,q i,1,q i,2,...,q i,li,q i+1,1 as a subpath. We then can apply the opeation M 1 again on H 1 and (q i 1,li 1,q i,1 ). It follows that M 1 (H 1, (q i 1,li 1,q i,1 ),m) with 1 m L i 1 1 is a Hamiltonian cycle of ExtC(G) 2 f. Theefoe, ExtC(G) 2 is 1-edge-Hamiltonian, and the lemma follows. Theoem 8. Let G be a tivalent 1-Hamiltonian gaph and C = x 0,x 1,x 2,...,x k 1,x 0 be a cycle of G. Then, ExtC(G) n is tivalent 1-Hamiltonian fo n 2. Poof. Since G is 1-Hamiltonian, it follows fom Lemma 4 and Coollay 3 that ExtC(G) 2 is 1-Hamiltonian and that OC 2 is ecoveable with espect to ExtC(G). 2 Assume that ExtC(G) k is 1-Hamiltonian and OC k is ecoveable with espect to ExtC(G) k fo 2 k l. Conside ExtC l+1 (G) =Ext O l C (ExtC(G)). l Then, by Theoem 7, Ext O l C (ExtC(G)) l = ExtC l+1 (G) is 1-Hamiltonian. Futhemoe, it follows fom Lemma 3 that OC l+1 is ecoveable with espect to ExtC l+1 (G), which makes the induction applicable. Hence, the theoem follows. Recusively applying cycle extensions, we can constuct a family of 1-Hamiltonian gaphs fom a known tivalent 1-Hamiltonian gaph. In what follows, we discuss two such families of gaphs, namely, eye netwoks and extended Petesen gaphs. The eye netwoks poposed by Wang et al. [6] ae, indeed, constucted by ecusive cycle extensions fom Eye(1) as shown in the next theoem. The 1-Hamiltonicity of Eye(n) is then a diect consequence of Theoem 8, wheeas in [6], the authos showed the 1-Hamiltonicity of Eye(n) using a diffeent appoach. Theoem 9. (i) Eye(n) =ExtO n 1 1 (Eye(1)) fo n 2, whee O 1 be the outemost cycle of Eye(1) (as shown by the dakened edges in Fig. 14). (ii) Eye(n) is 1-Hamiltonian fo n 3. Poof. It can be easily veified that Eye(2) is obtained by pefoming a cycle extension on Eye(1) aound O 1, that is, Eye(2) = Ext O1 (Eye(1)). Let O 2 be the outemost cycle of Eye(2). Note that O 2 is also the extended cycle of O 1 in Eye(2). Let O n denote the outemost cycle of Eye(n) fo n 1. It can be easily veified that fo n 3,Eye(n) =Ext On 1 (Eye(n 1)) and O n is the extended cycle of Ext On 1 (Eye(n 1)). Thus, Eye(n) =ExtO n 1 1 (Eye(1)) fo n 2. Note that Eye(1) is a 3-join of two K 4 and is also tivalent 1-Hamiltonian. It follows fom Theoem 8 that ExtO n 1 (Eye(1)), that is, Eye(n + 1), fo n 2, is 1- Hamiltonian. Hence, the theoem follows. By ecusively applying cycle extensions, we define extended Petesen gaphs, denoted by EP(n) fo n 0, as follows: (i) EP(0) is the Petesen gaph shown in Figue 17(a) with a specific cycle C indicated by the dakened edges. Define O 0 C = C. FIG. 17. Extended Petesen gaphs. 244 NETWORKS 2000

13 (ii) Define EP(1) = Ext O 0 C (EP(0)) as shown in Figue 17(b) and OC 1 = O C shown by the dakened edges in this figue. (iii) Fo n 2, define EP(n) =Ext O n 1 C On C as the extended cycle of OC n 1 in EP(n). Equivalently, we wite EP(n) =ExtC(EP(0)). n It is known that the Petesen gaph is 1-vetex- Hamiltonian but neithe Hamiltonian no 1-edge- Hamiltonian. We can use the popeties of cycle extensions to show the 1-Hamiltonicity of EP(n) as stated in the following theoem: Theoem 10. EP(n) is 1-Hamiltonian fo n 1. Poof. Since the Petesen gaph EP(0) is 1-vetex- Hamiltonian, it follows fom Lemma 1 that EP(1) is also 1-vetex-Hamiltonian. Futhemoe, EP(1) is Hamiltonian, whee a Hamiltonian cycle of EP(1) is illustated in Figue 17(c) by the dakened edges. Using pope otation of EP(1), we can always obtain a Hamiltonian cycle without containing any specific edge. It follows that EP(1) is 1-edge-Hamiltonian and, thus, 1-Hamiltonian. It can also be easily veified that OC 1 is ecoveable with espect to EP(1). It follows fom Theoems 7 and 8 that EP(n) is 1-Hamiltonian fo n CONCLUDING REMARKS In this pape, we intoduced thee constuction schemes, namely, 3-join, (3, 4)-join, and cycle extension, to constuct families of 1-Hamiltonian gaphs including seveal families aleady known in the liteatue and some new families. It follows fom Coollay 1 that tivalent 1- edge-hamiltonian gaphs ae closed unde 3-joins. It follows fom Lemma 1 that tivalent 1-vetex-Hamiltonian gaphs ae closed unde cycle extensions. Howeve, some families of 1-Hamiltonian gaphs can be geneated fom pefoming 3-joins on 1-edge-Hamiltonian gaphs and pefoming cycle extensions on 1-vetex-Hamiltonian gaphs. Fo example, the 1-Hamiltonian gaph shown in Figue 2(b) is obtained by pefoming a sequence of 3- joins on K 4 and K 3,3, whee K 3,3 is 1-edge-Hamiltonian but not 1-vetex-Hamiltonian. In this pape, 3-joins and cycle extensions on tivalent gaphs wee studied, although these opeations can be applied to abitay gaphs. We wonde whethe these opeations can be applied to nontivalent gaphs and still peseve the good popeties of the oiginal gaphs. On the othe hand, we want to exploe in the futue what popeties of gaphs can be peseved unde these opeations. REFERENCES [1] F. Haay and J.P. Hayes, Edge fault toleance in gaphs, Netwoks 23 (1993), [2] F. Haay and J.P. Hayes, Node fault toleance in gaphs, Netwoks 27 (1996), [3] C.N. Hung, L.H. Hsu, and T.Y. Sung, Chistmas tee: A vesatile 1-fault toleant design fo token ings, Info Pocess Lett 72 (1999), [4] K. Mukhopadhyaya and B.P. Sinha, Hamiltonian gaphs with minimum numbe of edges fo fault-toleant topologies, Info Pocess Lett 44 (1992), [5] J.J. Wang, C.N. Hung, and L.H. Hsu, Optimal 1- Hamiltonian gaphs, Info Pocess Lett 65 (1998), [6] J.J. Wang, T.Y. Sung, L.H. Hsu, and M.Y. Lin, A new family of optimal 1-Hamiltonian gaphs with small diamete, Poc 4th Annual Int Computing and Combinatoics Conf, Lectue Notes in Compute Science, Vol. 1449, Spinge- Velag, Belin, 1998, pp NETWORKS