On the Enhanced Hyper-hamiltonian Laceability of Hypercubes

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1 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Isse 4, Volme 2, 2008 On the Enhanced He-hamiltonian Laceabilit of Hecbes Tsng-Han Tsai, T-Liang Kng, Jimm J. M. Tan, and Lih-Hsing Hs Abstact A biatite gah is hamiltonian laceable if thee eists a hamiltonian ath beteen an to etices that ae in diffeent atite sets. A hamiltonian laceable gah G is said to be hehamiltonian laceable if, fo an ete of G, thee eists a hamiltonian ath of G {} oining an to etices that ae located in the same atite set diffeent fom that of. In this ae, e fthe imoe the he-hamiltonian laceabilit of hecbes b shoing that, fo an to etices, fom one atite set of Q n, n 4, and an ete fom the othe atite set, thee eists a hamiltonian ath H of Q n {} oining to sch that d H (, ) = l fo an ete V (Q {,, } and fo ee intege l satisfing both d Qn (, ) l 2 n 2 d Qn (, ) and 2 (l d Qn (, )). As a conseence, man attactie oeties of hecbes follo diectl fom o eslt. Keods Path embedding, Hamiltonian laceable, Hehamiltonian laceable, Inteconnection netok, Hecbe I. INTRODUCTION A mltiocesso/mlticomte inteconnection netok is sall modeled as a gah [2], [4], [8], [14], [16], [19], [20], in hich etices coesond to ocessos/comtes, and edges coesond to connections o commnication links. Thoghot this ae, a netok is eesented as a looless ndiected gah. Fo gah definitions and notations, e follo the ones gien b Hs and Lin [4]. A gah G is a to-tle (V, E), hee V is a nonemt set, and E is a sbset of {(, ) (, ) is an nodeed ai of V }. We sa that V is the ete set and E is the edge set. Fo conenience, e se V (G) and E(G) to denote the ete set and the edge set of G, esectiel. A gah G is biatite if its ete set V (G) is the nion of to disoint sbsets, denoted b V 0 (G) and V 1 (G), sch that ee edge oins a ete of V 0 (G) to a ete of V 1 (G). To etices, and, of a gah G ae adacent if (, ) E(G). A ath P of length k fom ete to ete in a gah G is a seence of distinct etices 1, 2,..., k+1 sch that 1 =, k+1 =, and Manscit eceied Jana, This eseach as atiall soted b the National Science Concil of the Reblic of China nde Contact NSC E MY3, and in at b the Aiming fo the To Uniesit and Elite Reseach Cente Deeloment Plan. Tsng-Han Tsai is ith the Deatment of Comte Science, National Chiao Tng Uniesit, 1001 Uniesit Rd., Hsinch, Taian ( tsaich@cs.nct.ed.t). T-Liang Kng is ith the Deatment of Comte Science, National Chiao Tng Uniesit, 1001 Uniesit Rd., Hsinch, Taian ( tlkeng@cs.nct.ed.t). Jimm J. M. Tan is a Pofesso in the Deatment of Comte Science, National Chiao Tng Uniesit, 1001 Uniesit Rd., Hsinch, Taian ( mtan@cs.nct.ed.t). Lih-Hsing Hs is a Pofesso in the Deatment of Comte Science and Infomation Engineeing, Poidence Uniesit, 200 Chng Chi Rd., Taichng, Taian (coesonding atho to oide hone: Et /18209; lhhs@cs..ed.t). ( i, i+1 ) E(G) fo ee 1 i k if k 1. Moeoe, a ath of length eo, consisting of a single ete, is denoted b. We also ite P as, P, to emhasie its beginning and ending etices. Fo conenience, e ite P as 1,..., i, Q,,..., k+1, hee Q = i,...,. In aticla, let P 1 = k+1, k,..., 1 denote the eese of P. A ccle is a ath ith at least thee etices sch that the last ete is adacent to the fist one. Fo clait, a ccle of length k is eesented b 1, 2,..., k, 1. The length of a ath P, denoted b l(p ), is the nmbe of edges in P. The distance beteen to distinct etices and in gah G, denoted b d G (, ), is the length of the shotest ath beteen and. A hamiltonian ccle (o hamiltonian ath) of a gah G is a ccle (o ath) that sans G. A biatite gah is hamiltonian laceable [15] if thee eists a hamiltonian ath beteen an to etices that ae in diffeent atite sets. A hamiltonian laceable gah G is he-hamiltonian laceable [9] if, fo i {0, 1} and fo an ete V i (G), thee eists a hamiltonian ath of G {} beteen an to etices of V 1 i (G). The hecbe is one of the most ola inteconnection netoks fo aallel comte/commnication sstem [4], [8], [19]. This is atl de to its attactie oeties sch as eglait, ecsie stcte, ete and edge smmet, maimm connectiit, as ell as effectie oting and boadcasting algoithm. The definition of hecbes is esented as follos. Fo clait, e se boldface lettes to denote n- bit bina stings. Let = b n 1... b i... b 0 be an n-bit bina sting. Fo an i, 0 i n 1, e se () i to denote the bina sting b n 1... b i... b 0. Moeoe, e se () i to denote the bit b i of. The Hamming eight of, denoted b H (), is defined as {0 n 1 () = 1}. The n-dimensional hecbe (o n-cbe fo shot) Q n consists of 2 n etices and n2 n 1 edges. Each ete coesonds to an n-bit bina sting. To etices and ae adacent if and onl if = () i fo some i, and e call the edge (, () i ) i-dimensional. The Hamming distance beteen and, denoted b h(, ), is defined to be {0 n 1 () () }. Hence, to etices and ae adacent if and onl if h(, ) = 1. Cleal, d Qn (, ) eals h(, ), and Q n is a biatite gah ith atite sets V 0 (Q = { V (Q H () is een} and V 1 (Q = { V (Q H () is odd}. Moeoe, Q n is ete-smmetic and edge-smmetic [8]. The oblem of embedding aths into hecbe is idel addessed b man eseaches [1], [3], [5], [6], [10], [12], [13], [18]. In aticla, Li et al. [10] oed that beteen an to diffeent etices and of Q n thee eists a ath 95

2 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Isse 4, Volme 2, 2008 P l (, ) of length l fo an l ith h(, ) l 2 n 1 and 2 (l h(, )). It is intiging to conside hethe sch ath P l (, ) can be fthe etended b inclding the emaining etices not in P l (, ) to fom a hamiltonian ath beteen and some othe ete. Fo this motiation, Lee et al. [7] shoed that, fo an to etices and fom diffeent atite set of Q n and fo an ete V (Q {, }, thee eists a hamiltonian ath R fom to sch that d R (, ) = l fo an intege l satisfing h(, ) l 2 n 1 h(, ) and 2 (l h(, )). In this ae, e fthe imoe sch eslt b oing that, fo an to etices, fom one atite set of Q n and an ete fom the othe atite set, thee eists a hamiltonian ath H beteen and in Q n {} sch that d H (, ) = l fo an ete V (Q {,, } and fo ee intege l satisfing both d Qn (, ) l 2 n 2 d Qn (, ) and 2 (l d Qn (, )). Accoding to the oosed imoement, man attactie oeties of hecbes follo diectl fom o eslt. II. PRELIMINARIES Lemma 1-4 and Theoem 5 ee oed in [7], [9], [15], [17]. The ill be sed to oe o main eslt in the net section. Lemma 1: [15] Fo an ositie intege n, Q n is hamiltonian laceable. Lemma 2: [9] Fo an ositie intege n, Q n is hehamiltonian laceable. Lemma 3: [17] Sose that n 4. If and ae an to etices fom diffeent sets of Q n, then Q n {, } is hamiltonian laceable. To aths P 1 and P 2 ae disoint if V (P 1 ) V (P 2 ) =. The folloing lemma is oed b Lee et al. [7]. Lemma 4: [7] Assme that n 4. Let, be an ai of distinct etices in V 0 (Q. Moeoe, let, be an ai of distinct etices in V 1 (Q. Sose that l 1 and l 2 ae to abita odd integes satisfing l 1 h(, ), l 2 h(, ), and l 1 + l 2 = 2 n 2. Then thee eist to disoint aths P 1 and P 2 sch that the folloing to conditions ae satisfied: (1) P 1 oins to ith l(p 1 ) = l 1, and (2) P 2 oins to ith l(p 2 ) = l 2. No, e make some emaks to illstate that some inteesting oeties of hecbes ae conseences of Lemma 4. Remak 5: The hamiltonian laceable oet of hecbes, oed in [15], states that thee eists a hamiltonian ath of Q n oining an ete V 0 (Q to an ete V 1 (Q. No, e oe that Q n is hamiltonian laceable b Lemma 4. Obiosl, Q n is hamiltonian laceable fo n = 1, 2, 3. Since n 4, e can choose a ai of adacent etices and sch that V 1 (Q ith and V 0 (Q ith. B Lemma 4, thee ae to disoint aths P 1 and P 2 sch that (1) P 1 is a ath oining to, (2) P 2 is a ath oining to, and (3) P 1 P 2 sans Q n. Obiosl,, P 1,,, P 2, foms a hamiltonian ath oining to. Ths, Q n is hamiltonian laceable. Remak 6: The bianconnected oet of Q n, oed in [10], stated that beteen an to diffeent etices and of Q n thee eists a ath P l (, ) of length l fo an l ith h(, ) l 2 n 1 and 2 (l h(, )). No, e oe that Q n is bianconnected b Lemma 4. Obiosl, Q n is bianconnected fo n = 1, 2, 3. No, e conside n 4. Withot loss of genealit, e assme that V 0 (Q. Sose that V 1 (Q. Ths, h(, ) is odd. Let l be an odd intege ith h(, ) l 2 n 1. Sose that l = 2 n 1. B Remak 5, Q n is hamiltonian laceable. Obiosl, the hamiltonian ath of Q n oining and is of length 2 n 1. Sose that l < 2 n 1. Since n 4, e can choose a ai of adacent etices and sch that V 0 (Q ith and V 0 (Q ith. Obiosl, h(, ) = 1. B Lemma 4, thee eist to disoint aths P 1 and P 2 sch that (1) P 1 is a ath oining to ith l(p 1 ) = 2 n 2 l, (2) P 2 is a ath oining to ith l(p 2 ) = l, and (3) P 1 P 2 sans Q n. Obiosl, P 2 is a ath of length l oining to. Sose that V 0 (Q. Ths, h(, ) is een. Let l be an een intege ith h(, ) l < 2 n 1. Since n 4, e can choose to diffeent neighbos and of sch that h(, ) = h(, ) 1. B Lemma 4, thee eist to disoint aths P 1 and P 2 sch that (1) P 1 is a ath oining to ith l(p 1 ) = l 1, (2) P 2 is a ath oining to ith l(p 2 ) = 2 n l 1, and (3) P 1 P 2 sans Q n. Obiosl,, P 1,, is a ath of length l oining to. Ths, Q n is bianconnected. Remak 7: The edge-biancclic oet oet of Q n, oed in [10], stated that fo an edge e = (, ) and fo an een intege ith 4 l 2 n thee eists a ccle of length l containing the edge e if n 2. Again, e e oe that Q n is edge-biancclic b Lemma 4. Obiosl, Q n is edgebiancclic fo n = 2, 3. Ths, e conside n 4. Sose that l = 2 n. B Remak 5, thee eists a hamiltonian ath P oining to. Obiosl,, P,, foms a hamiltonian ccle of length 2 n containing the edge e. Sose that l < 2 n. Since n 4, e can choose a ai of adacent etices and sch that V 0 (Q. B Lemma 4, thee eist to disoint aths P 1 and P 2 sch that (1) P 1 is a ath oining to ith l(p 1 ) = 2 n l 1, (2) P 2 is a ath oining to ith l(p 2 ) = l 1, and (3) P 1 P 2 sans Q n. Obiosl,, P 2,, is a ccle of length l containing the edge e. Ths, Q n is edge-biancclic fo n 2. No, e stat to oe Lemma 4. Poof: B bte foce, e can check the theoem holds fo n = 4. Assme the theoem holds fo an Q k ith 4 k < n. Withot loss of genealit, e can assme that l 1 l 2. Ths, l 2 2 n 1 1. Since Q n is ete-smmetic and edge-smmetic, e can assme that V 0 (Q 0 n 1) and V 0 (Q 1 n 1). We hae the folloing cases. Case 1: V 1 (Q 0 n 1) and V 1 (Q 1 n 1). Sose that l 2 < 2 n 1 1. B Remak 5, thee eists a hamiltonian ath R of Q 0 n 1 oining and. Since the length of R is 2 n 1 1, e can ite R as, R 1,,, R 2, fo some ete V 1 (Q ith and some ete V 0 (Q ith. Obiosl, h(, = 1. B indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining to ith l(s 1 ) = l 1 2 n 1, (2) S 2 is a ath oining to ith l(s 2 ) = l 2, and (3) S 1 S 2 sans Q 1 n 1. We set P 1 as, R 1,,, S 1,,, R 2, and 96

3 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Isse 4, Volme 2, 2008 set P 2 as S 2. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(a) fo illstation. Sose that l 2 = 2 n 1 1. B Remak 5, thee eists a hamiltonian ath P 1 of Q 0 n 1 oining and and thee eists a hamiltonian ath P 2 of Q 1 n 1 oining and. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(b) fo illstation. Case 2: {, } V 1 (Q 1 n 1). Sose that l 2 < 2 n 1 1. We choose a neighbo of sch that. Obiosl, V 0 (Q. B indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining to ith l(s 1 ) = l 1 2 n 1, (2) S 2 is a ath oining to ith l(s 2 ) = l 2, and (3) S 1 S 2 sans Q 1 n 1. B Remak 5, thee eists a hamiltonian ath R of Q 0 n 1 oining and. We set P 1 as, R,,, S 1, and e set P 2 as S 2. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(c) fo illstation. Sose that l 2 = 2 n 1 1. Again, e choose a neighbo of sch that. B indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining to ith l(s 1 ) = 1, (2) S 2 is a ath oining to ith l(s 2 ) = 2 n 1 3, and (3) S 1 S 2 sans Q 1 n 1. Obiosl, e can ite S 2 as, S2, 1, s, S2, 2 fo some ete V 1 (Q ith n. Again b indction, thee eist to disoint aths R 1 and R 2 sch that (1) R 1 is a ath oining to ith l(r 1 ) = 2 n 1 3, (2) R 2 is a ath oining n to s n ith l(r 2 ) = 1, and (3) R 1 R 2 sans Q 0 n 1. We set P 1 as, R 1,,, and set P 2 as, S2, 1, n, s n, s, S2, 2. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(d) fo illstation. Case 3: V 1 (Q 0 n 1) and V 1 (Q 1 n 1). Sose that l 2 = 1. Obiosl, = n. Let be a neighbo of in Q 0 n 1 sch that n and let be a neighbo of in Q 0 n 1 sch that. B indction, thee eist to disoint aths R 1 and R 2 sch that (1) R 1 is a ath oining to and l(r 1 ) = 2 n 1 3, (2) R 2 is a ath oining to and l(r 2 ) = 1, and (3) R 1 R 2 sans Q 0 n 1. Obiosl, V 1 (Q and V 0 (Q. Again b indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining to ith l(s 1 ) = 2 n 1 3, (2) S 2 is a ath oining to ith l(s 2 ) = 1, and (3) S 1 S 2 sans Q 1 n 1. We set P 1 as, R 1,,,,, S 1, and set P 2 as,. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(e) fo illstation. Sose that l 2 3. We set be a neighbo in Q 0 n 1 of ith if h(, ) = 1 and set be a neighbo of in Q 0 n 1 ith and h(, ) = h(, ) 1 if h(, ) 3. Let be a neighbo n in Q 0 n 1 sch that and. Ths, h(, ) = 1. B indction, thee eist to disoint aths R 1 and R 2 sch that (1) R 1 is a ath oining to ith l(r 1 ) = 2 n 1 3, (2) R 2 is a ath oining to ith l(r 2 ) = 1, and (3) R 1 R 2 sans Q 0 n 1. Again b indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining to ith l(s 1 ) = l 1 2 n 1 + 2, (2) S 2 is a ath oining to ith l(s 2 ) = 1 2 2, and (3) S 1 S 2 sans Q 1 n 1. We set P 1 as, R 1,,, S 1, and set P 2 as, S 2,,,. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(f) fo illstation. Case 4: {, } V 1 (Q 0 n 1). Sose that l 2 = 1. Obiosl, = n. B Remak 5, thee eist a hamiltonian ath R of Q 0 n 1 oining to. Obiosl, R can be itten as, R 1,,,, R 2,. Note that = if l(r 1 ) = 0. Obiosl, and ae in V 0 (Q. Ths, and ae in V 1 (Q. Let be a neighbo of in Q 1 n 1 sch that. B indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining to ith l(s 1 ) = 2 n 1 3, (2) S 2 is a ath oining to ith l(s 2 ) = 1, and (3) S 1 S 2 sans Q 1 n 1. We set P 1 as, R 1,,, S 1,,,, R 2, and set P 2 as,. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(g) fo illstation. Sose that l 2 3. We set be a neighbo of in Q 0 n 1 ith if h(, ) = 1 and set be a neighbo of in Q 0 n 1 ith and h(, ) = h(, ) 1 if h(, ) 3. B indction, thee eist to disoint aths R 1 and R 2 sch that (1) R 1 is a ath oining to ith l(r 1 ) = 2 n 1 3, (2) R 2 is a ath oining to ith l(r 2 ) = 1, and (3) R 1 R 2 sans Q 0 n 1. Obiosl, e can ite R 1 as, R1, 1 s, t, R1, 2 fo some ete s V 1 (Q sch that s n. B indction, thee eist to disoint aths S 1 and S 2 sch that (1) S 1 is a ath oining s n to t n ith l(s 1 ) = l 1 2 n 1 2, (2) S 2 is a ath oining to ith l(s 2 ) = l 2 2, and (3) S 1 S 2 sans Q 1 n 1. We set P 1 as, R1, 1 s, s n, S 1, t n, t, R1, 2 and set P 2 as, S 2,,,. Obiosl, P 1 and P 2 ae the eied aths. See Fige 4(h) fo illstation. B Lemma 4, the net eslt can be deied. Theoem 8: [7] Assme that n 2. Let and be an to etices fom diffeent atite sets of Q n, and let be an ete of Q n {, }. Then thee eists a hamiltonian ath H of Q n oining to sch that d H (, ) = l fo an intege l satisfing both h(, ) l 2 n 1 h(, ) and 2 (l h(, )). Poof: B bte foce, e can check the theoem holds fo n = 2, 3. No, e conside n 4. Withot loss of genealit, e assme that V 0 (Q and V 1 (Q. Sose that V 1 (Q. Obiosl, h(, ) 2. Thee eists a neighbo of sch that and h(, ) = h(, ) 1. Obiosl, V 0 (Q. B Lemma 4, thee eist to disoint aths R 1 and R 2 sch that (1) R 1 is a ath oining to ith l(r 1 ) = l, (2) R 2 is a ath oining to ith l(r 2 ) = 2 n l 2, and (3) R 1 R 2 sans Q n. We set R as, R 1,,, R 2,. Obiosl, R is the eied hamiltonian ath. Sose that V 0 (Q. Obiosl, h(, ) 2. Thee eists a neighbo of sch that and h(, ) = h(, ) 1. Obiosl, V 1 (Q. B Lemma 4, thee eist to disoint aths R 1 and R 2 sch that (1) R 1 is a ath oining to ith l(r 1 ) = l 1, (2) R 2 is a ath oining to ith l(r 2 ) = 2 n l 1, and (3) R 1 R 2 sans Q n. We set R as, R 1,,, R 2,. Obiosl, R is the eied hamiltonian ath. III. HYPER-HAMILTONIAN LACEABILITY Fo 0 n 1 and i {0, 1}, let Q,i n be a sbgah of Q n indced b { V (Q () = i}. Obiosl, Q,i n is 97

4 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Isse 4, Volme 2, 2008 n s n s (a) (b) (c) (d) t s t n s n (e) (f) (g) (h) Fig. 1. Illstation fo Lemma 4. isomohic to Q n 1. Then e can oe o main eslt b indction. Theoem 9: Sose that n 4. Let, be an ai of distinct etices in V 1 (Q and be an ete of V 0 (Q. Moeoe, let V (Q {,, }. Then Q n {} contains a hamiltonian ath H beteen and sch that d H (, ) = l fo ee intege l satisfing both d Qn (, ) l 2 n 2 d Qn (, ) and 2 (l d Qn (, )). Poof: We oe this theoem b indction on n. As the indction basis, e check, b comte ogam, that this eslt holds fo n = 4. Assme that the eslt is te fo an intege k ith 4 k < n. Since Q n is ete-smmetic and edge-smmetic, e assme that = 0 n. If V 0 (Q, then let be an intege of {0, 1,..., n 1} sch that () () and (). Otheise, let be an intege of {0, 1,..., n 1} sch that () () and (). Then e can atition Q n along dimension into Q,0 n and Q,1 n. Withot loss of genealit, e assme that is in Q,0 n, and is in Q,1 n. To constct a hamiltonian ath H of Q n {} oining to ith d H (, ) = l, the folloing cases ae distingished. Case 1: Sose that V 0 (Q,0. We fthe conside the folloing sbcases. Sbcase 1.1: Sose that l 2 n 1 3. Let be a ete of Q,0 n adacent to. B indctie hothesis, Q,0 n {} has a hamiltonian ath L beteen and sch that d L (, ) = l fo ee intege l satisfing both d Qn (, ) l 2 n 1 2 d Qn (, ) = 2 n 1 3 and 2 (l d Qn (, )). Ths e can ite L as, L 1,, L 2,. B Lemma 1, Q,1 n is hamiltonian laceable. Since () and ae in the diffeent atite sets, Q,1 n has a hamiltonian ath R beteen () and. Let H =, L 1,, L 2,, (), R,. As a eslt, H is a hamiltonian ath of Q n {} oining to sch that d H (, ) = l fo an odd intege l 2 n 1 3. See Fige 2(a) fo illstation. Sbcase 1.2: Sose that l 2 n 1 1. Let k {0, 1,..., n 1} {} sch that () k () k. Then e set to be () k. Tiiall e hae d Qn (, ) = 1. Fistl, e conside that (). B indctie hothesis, Q,0 n {} has a hamiltonian ath L beteen and sch that d L (, ) = 1. Accodingl, e can ite L as, L 1,,. Since l(l 1 ) = 2 n fo n 4, e ite L 1 as, L 1,,, hee is some ete of Q,0 n adacent to. Hence ath L can be eesented as, L 1,,,. B Lemma 4, thee eist to disoint aths R 1 and R 2 in Q,1 n sch that R 1 oins () to () ith l(r 1 ) = l 2 n 1 + 2, and R 2 oins () to ith l(r 2 ) = 2 n l 4. Let H =, L 1,, (), R 1, (),,, (), R 2,. As a eslt, H tns ot to be a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l fo an odd intege l 2 n 1 1 if (). See Fige 2(b) fo illstation. Secondl, e conside that () =. Again, the indctie hothesis enses that Q,0 n {} has a hamiltonian ath L beteen and sch that d L (, ) = 1. Hence e can ite L as, L 1,,. Since l(l 1 ) = 2 n fo n 4, e ite L 1 as,, L 1,, hee is some ete of Q,0 n adacent to. Accodingl, ath L can be eesented as,, L 1,,. B Lemma 4, thee eist to disoint aths R 1 and R 2 in Q,1 n sch that R 1 is a ath of length l 2 n oining () to (), and R 2 is a ath of length 2 n l 4 oining () to. Let H =, (), R 1, (),, L 1,,, (), R 2,. As a conseence, H foms a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l if () =. See Fige 2(c) fo illstation. Case 2: Sose that V 1 (Q,0. Simila to the case descibed ealie, e conside the folloing sbcases. Sbcase 2.1: Sose that l 2 n 1 4. Let V 1 (Q,0 sch that d Qn (, ) = 2. B indctie hothesis, Q,0 n {} has a hamiltonian ath L beteen and sch that d L (, ) = l fo ee intege l satisfing both d Qn (, ) l 2 n 1 2 d Qn (, ) = 2 n 1 4 and 2 (l d Qn (, )). Fo clait, e ite L as, L 1,, L 2,. Since () and ae in the diffeent atite sets of Q n, Lemma 1 enses that Q,1 n has a hamiltonian ath R beteen () and. Let H =, L 1,, L 2,, (), R,. Conseentl, ath H tns 98

5 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Isse 4, Volme 2, 2008 () () () () () () () () = (a) (b) (c) Fig. 2. V 0 (Q,0. (a) l 2 n 1 3. (b) l 2 n 1 1 and (). (c) l 2 n 1 1 and () =. () () () () () (a) (b) (c) Fig. 3. V 1 (Q,0. (a) l 2 n 1 4. (b) l 2 n 1. (c) l = 2 n 1 2. ot to be a hamiltonian ath of Q n {} oining and sch that d H (, ) = l fo an een intege l 2 n 1 4. See Fige 3(a) fo illstation. Sbcase 2.2: Sose that l 2 n 1. Let be a ete of Q,0 n sch that h(, ) = 1 and (). B indctie hothesis, Q,0 n {} has a hamiltonian ath L beteen and sch that d L (, ) = 1. Fo clait, e ite L as,, L 1,. B Lemma 4, thee eist to disoint aths R 1 and R 2 sch that R 1 is a ath of length l 2 n oining () to (), and R 2 is a ath of length 2 n l 3 oining () to. Let H =, (), R 1, (),, L 1,, (), R 2,. Theefoe, H is a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l. See Fige 3(b) fo illstation. Sbcase 2.3: Sose that l = 2 n 1 2. Since and ae in the same atite set, Lemma 2 enses that Q,0 n {} has a hamiltonian ath L beteen and. B Lemma 1, Q,1 n is hamiltonian laceable. Ths Q,1 n has a hamiltonian ath R beteen () and. Let H =, L,, (), R,. As a eslt, H is a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l = 2 n 1 2. See Fige 3(c) fo illstation. Case 3: Sose that V 0 (Q,1. We conside the folloing sbcases. Sbcase 3.1: Sose that l 2 n 1 3. Let be a ete of Q,0 n sch that d Qn (, () ) = 2. B indctie hothesis, Q,0 n {} has a hamiltonian ath L beteen and sch that d L ((), ) = l 1. Fo clait, e ite L as, L 1, (), L 2,. Since d Qn (, () ) = 2, ath L 2 can be itten as (),, L 3,, hee is some ete of Q,0 n adacent to (). Fistl, e assme that (). B Lemma 4, thee eist to disoint aths R 1 and R 2 sch that R 1 oins to (), R 2 oins () to, and l(r 1 ) + l(r 2 ) = 2 n 1 2. Let H =, L 1, (),, R 1, (),, L 3,, (), R 2,. Then H is a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l. See Fige 4(a) fo illstation. Secondl, e assme that () =. B Lemma 2, Q,1 n {} has a hamiltonian ath R beteen and (). Let H =, L 1, (),, R, (),, L 1 3,,. Then H foms a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l. See Fige 4(b) fo illstation. Sbcase 3.2: Sose that l 2 n Let V 0 (Q,0 sch that d Qn (, ) = 1 and (). B indctie hothesis, Q,0 n {} has a hamiltonian ath L beteen and () sch that d Qn (, ) = 1. Ths e can ite L as,, L 1, (). B Lemma 4, thee eist to disoint aths R 1 and R 2 sch that R 1 oins () to () ith l(r 1 ) = l 2 n 1 and R 2 oins to ith l(r 2 ) = 2 n l 2. Let H =, (), R 1, (),, L 1, (),, R 2,. Then H is a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l. See Fige 4(c) fo illstation. Sbcase 3.3: Sose that l = 2 n 1 1. B Lemma 2, Q,0 n is he-hamiltonian laceable. Since and () belong to the same atite set of Q n, Q,0 n {} has a hamiltonian ath L beteen and (). B Lemma 1, Q,1 n is hamiltonian 99

6 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Isse 4, Volme 2, 2008 () () () () () (a) (b) () () () () (c) (d) Fig. 4. V 0 (Q,1. (a) l 2 n 1 3 and (). (b) l 2 n 1 3 and () =. (c) l 2 n (d) l = 2 n 1 1. laceable. Theefoe, Q,1 n has a hamiltonian ath R beteen and. We set H to be, L, (),, R,. As a eslt, H is a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l = 2 n 1 1. See Fige 4(d) fo illstation. Case 4: Sose that V 1 (Q,1. Similal, e conside the folloing sbcases. Sbcase 4.1: Sose that l 2 n 1 2. Let V (Q,1 sch that d Qn (, ) = 1 and (). B Lemma 3, Q,0 n {, } is hamiltonian laceable. Cleal, () and () ae in diffeent atite sets. Theefoe, Q,0 n {, } has a hamiltonian ath L beteen () and (). B Lemma 4, thee eist to disoint aths R 1 and R 2 sch that R 1 oins () to ith l(r 1 ) = l 1 and R 2 oins () to ith l(r 2 ) = 2 n 1 l 1. Then e set H to be, (), R 1,, (), L, (),, R 2,. As a eslt, H is a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l. See Fige 5(a) fo illstation. Sbcase 4.2: Sose that l 2 n 1. Let V (Q,0 sch that d Qn ((), ) = 1. B Lemma 2, Q,0 n {} has a hamiltonian ath L beteen and. B Theoem 8, thee eists a hamiltonian ath R of Q,1 n oining () to sch that d R ((), ) = l 2 n Then e set H to be, L,, (),, R,. Conseentl, H foms a hamiltonian ath of Q n {} beteen and sch that d H (, ) = l. See Fige 5(b) fo illstation. IV. CONCLUSION In this ae, e oe that, fo an to etices and fom one atite set of Q n (n 4) and fo an ete fom the othe atite set, thee eists a hamiltonian ath H of Q n {} oining to sch that d H (, ) = l fo an ete V (Q {,, } and fo ee intege l satisfing both d Qn (, ) l 2 n 2 d Qn (, ) and 2 (l d Qn (, )). Secificall, e gie the folloing eamle to indicate h sch a eslt is not te fo Q 3. See Fige 6 fo illstation, in hich e assme that = 100, = 010, = 000, and = 110. Cleal, thee does not eist an hamiltonian ath H in Q 3 {} oining to sch that d H (, ) = Fig. 6. Q 3 {} has no hamiltonian ath H beteen and sch that d H (, ) = REFERENCES [1] X.-B. Chen, Ccles assing thogh escibed edges in a hecbe ith some falt edges, Inf. Pocess. Lett., 104, 2007, [2] S.-Y. Chen and S.-S. Kao, The edge-ancclicit of Dal-cbe Etensie Netoks, Poceedings of the 2nd WSEAS Int. Conf on COMPUTER EN- GINEERING and APPLICATIONS (CEA 08), Acalco, Meico, Jana 25-27, 2008,

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