Matchings in Cayley Graphs of S n. North Carolina State University, Box Abstract

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1 Hamilon Cycle which Exend Tranpoiion Maching in Cayley Graph of S n Frank Rukey Deparmen of Compuer Science Unieriy of Vicoria, P. O. Box 1700 Vicoria, B. C. V8W 2Y2 CANADA frukey@cr.uic.ca Carla Saage y Deparmen of Compuer Science Norh Carolina Sae Unieriy, Box 8206 Raleigh, NC cd@ccadm.ncu.edu Abrac Le B be a bai of ranpoiion for S n and le Cay(B :S n )behecayley graph of S n wih repec o B. Iwa hown by Kompel'makher and Likoe ha Cay(B :S n ) i hamilonian. We exend hi reul a follow. Noe ha eery ranpoiion b in B induce a perfec maching M b in Cay(B :S n ). We how here when n>4 ha for any b 2 B, here i a Hamilon cycle in Cay(B :S n ) which include eery edge of M b. Tha i, for n>4, for any baib of ranpoiion of S n, and for any b 2 B, i i poible o generae all permuaion of 1 2 ::: n by ranpoiion in B o ha eery oher ranpoiion i b. Keyword. Cayley graph, perfec maching, hamilonian graph, ranpoiion. AMS(MOS) ubec claicaion. 05C25, 05C45. Reearch uppored by he Naural Science and Engineering Reearch Council of Canada under gran A3379. y Reearch uppored by he Naional Science Foundaion Gran No. CCR

2 1 Inroducion For a nie group G wih generaing e X, he Cayley graph of G wih repec o he generaing e X i he graph Cay(X :G) wiherex e G, in which g and gx are oined by an undireced edge for eery g 2 G and x 2 X. We will hink of he edge fg gxg a being labeled x. A compelling queion in graph heory i wheher eery undireced Cayley graph i hamilonian. Alhough here are reul uch a [CuWi] and [KeWi] which how ha he anwer i ye for cerain ubclae of Cayley graph, he general queion remain open. If we require only a Hamilon pah, he queion i ill open and i, in fac, a pecial cae of he more general conecure of Loaz ha eery conneced, undireced, erex raniie graph ha a Hamilon pah [Lo]. If we reric our aenion o he cae when G = S n, he ymmeric group of all permuaion of [n] =f1 2 ::: ng, i i ill an open problem wheher eery Cayley graph of S n i hamilonian. The queion remain open een when we require ha eery generaor x 2 X aify x 2 = id. Whaiknown i ha for eery generaing e X of ranpoiion, hecayley graph of S n i hamilonian. Thi wa r hown by Kompel'makher and Likoe [KoLi]. Slaer howed in [Sl] ha one could alway nd a Hamilon pah in Cay(X :S n ) which ended a a permuaion wih a in poiion k for any k 2 [n]. Tchuene generalized boh of hee reul by howing ha any wo permuaion of dieren pariy are oined by a Hamilon pah in Cay(X :S n ) [Tc]. A an example, he well-known algorihm of Seinhau [S], ohnon [o], and Troer [Tr], for generaing permuaion by adacen ranpoiion, gie a Hamilon cycle hrough he Cayley graph of S n wih generaing e f(12) (23) (34) ::: (n;1 n)g. Howeer, an elemen ofs n of order wo need no be a ranpoiion, o i remain open wheher he Cayley graph of S n on a e of generaor, each of order wo, i hamilonian. Recenly i ha been hown ha he Cayley graph of Coxeer group, generaed by order wo elemen which are geomeric reecion, are hamilonian [CoSlWi]. A relaed reul i ha for A n generaed by he e of 3-cycle f(12n) (13n) ::: (1 n;1 n)g, hecayley graph i hamilonian [GoRo]. 2

3 In hi paper, we conider S n wih any generaing e of ranpoiion, X. Noe ha each x 2 X dene a perfec maching in Cay(X :S n ), ha i, a e M x of edge of he graph wih he propery haeacherex of Cay(X :S n ) i he end of exacly one edge in M x : M x = ffg gxgg 2 S n g: Knowing ha Cay(X :S n ) i hamilonian by [KoLi], we can ak if M x exend o a Hamilon cycle. Such a cycle correpond o a liing of all permuaion of [n], in which ucceie permuaion dier by a ranpoiion in X, o ha alernae ranpoiion correpond o he elemen x. The graph C = Cay(f(12) (23) (34)g:S 4 )ihown in Figure 1. The ripled line of he gure indicae edge in he maching M (23), and he li of permuaion of Figure 2 i a Hamilon cycle in C ha conain eery edge of M (23). A pecic inance of hi problem aroe iniially in he work of Pruee and Rukey on liing he linear exenion of cerain poe by ranpoiion [PrRu]. Le R be he cla of ranked poe in which eery non-maximal elemen ha a lea wo upper coer. Example of poe in R include he odd fence, crown, he Boolean algebra laice, he laice of ubpace of a nie-dimenional ecor pace oer GF (q), and pariion laice. In [PrRu] i i proen ha he linear exenion of any poe in R can be lied o ha eery exenion dier by a ranpoiion from i predeceor in he li. Their proof required a cyclic liing of all permuaion of [n] by ranpoiion o ha eery oher ranpoiion wa an exchange of he elemen in poiion 1 and 2. Alhough hey were able o how uchaliingwa alway poible, in ome cae he ranpoiion were no of elemen in adacen poiion hee ranpoiion were he only one in he proof ha were nonadacen. In [RuSa] we howed ha i i poible o li permuaion of [n] by adacen ranpoiion o ha eery oher ranpoiion exchange he elemen in poiion 1 and 2. See Figure 3 for an example when n =5. Thi reul i equialen o howing ha in he Cayley 3

4 QQ Q Q Q QQ Q Q Q Figure 1: The graph Cay(f(12) (23) (34)g : S 4 ) wih M (23) b Figure 2: B = f(12) (23) (34)g and b = (23) (read acro.) 4

5 b Figure 3: B = f(12) (23) (34) (45)g and b = (12) (read acro.) graph Cay(X :S n ) where X = f(12) (23) ::: (n;1 n)g, he perfec maching M (12) exend o a Hamilon cycle. For n = 4 here i a Hamilon pah including eery edge of M (12), bu no Hamilon cycle. A conequence of hi reul, which i a pecial cae of our main heorem below, i ha he linear exenion of he poe in R can, in fac, be lied by adacen ranpoiion. Our maor reul in hi paper i he following heorem. Main Theorem. LeX be a generaing e of ranpoiion for S n,wheren>4. Then for any x 2 X, M x exend o a Hamilon cycle in Cay(X :S n ). A bai for S n i a minimal e of generaor for S n. Wihou lo of generaliy, we may aume ha our generaing e of ranpoiion for S n i a bai, call i B, o ha he ranpoiion can be decribed a a ree T B : he erice of T B are he poiion 1 2 ::: n, where i and are oined by an edge if and only if (i) ia 5

6 ; ;; 6 ; ; ;; 7 ; ;; Figure 4: Excepional combinaion: ar (lef) and are (righ) wih b a indicaed. ranpoiion in B. For b 2 B we refer o he ordered pair ht B bi a a combinaion. A combinaion ht B bi i aid o be ordinary if here are wo edge e 1 e 2 in T B uch ha (a) e 1 6= b, e 2 6= b, and (b) he edge e 1 and e 2 are no adacen. A combinaion ha i no ordinary i excepional. The reaon for diinguihing beween ordinary and excepional combinaion i ha our baic proof echnique i o plice ogeher Hamilon cycle in cerain induced ubgraph. Thi plicing i baed on mall cycle ha don' conain any edge labeled b. IfhT B bi i ordinary hen eery erex of Cay(B :S n ) i on a 4-cycle wih no edge labeled b. Specically, ifc d 6= b are nonadacen edgeoft B hen (cd) 2 = id, o for any erex of Cay(B :S n ), he equence c cd cdc cdcd = i a 4-cycle. Howeer, if ht B bi i excepional, any edgec d 6= b of T B are adacen, o generaor c and d do no commue. In hi cae here will be no 4-cycle in Cay(B :S n ) no conaining b. Inead, (cd) 3 = id, which gie rie o 6-cycle no conaining b. A ar i a ree of n erice in which oneerex ha degree n ; 1 and a are i a ree of n erice in which oneerex ha degree n ; 2andoneerex ha degree 6

7 2. We refer o a erex of degree one in a ree a a leaf of he ree. See Figure 4. Excepional combinaion are characerized in he following lemma, which we ae wihou proof. Lemma 1 An excepional combinaion ht (i)i for n>4 mu eiher be a ar or be a are in which i i a leaf and i a erex of degree 2 (or ice-era). The proof of he Main Theorem ue a dieren conrucion for each ofhe following hree familie of combinaion: 1. Ordinary combinaion. 2. Excepional combinaion in which n > 4andT i a are. 3. Excepional combinaion in which n > 4andT i a ar. Wihin each family, he conrucion relie induciely only on member of he ame family, o he hree cae can be handled independenly. Secion 2 concern ordinary combinaion. Excepional combinaion are handled in Secion 3. Secion 4 conain exenion and open problem. 2 Ordinary Combinaion An ordinary combinaion ht (i)i i minimal if for eery leaf k 6= i he combinaion ht ; k (i)i i excepional. The following lemma ieailyproen. Lemma 2 There are hree non-iomorphic minimal ordinary combinaion. They are hown in Figure 2, 3, 5. For b 2 B, dene a b-alernaing pah (cycle) o be a pah (cycle) in Cay(B :S n ) in which alernae edge are labeled b. Furhermore, in he cae of a b-alernaing pah, he r and la edge of he pah mu be labeled b. For example, he cycle in Figure 3 i a (12)-alernaing Hamilon cycle in Cay(B :S n )whereb = f(12) (23) ::: (n;1 n)g. 7

8 b Figure 5: B = f(12) (23) (24) (45)g and b = (12) (read acro.) In hi ecion we will how ha when B i a bai of ranpoiion for S n, wih b 2 B and ht B bi i an ordinary combinaion, hen Cay(B :S n ) ha a b-alernaing Hamilon cycle. The proof i by an inducie conrucion and will require a omewha ronger hypohei. If Q i any b-alernaing cycle in Cay(B :S n ), an (i )-inerion pair for Q i a pair of conecuie erice, on Q aifying (1) (i) =(i)= and (2) he edge oining and i no labeled b (i.e., f g 62 M b ). Theorem 1 Le B be a bai of ranpoiion for S n, and le b 2 B be uch ha ht B bi i an ordinary combinaion. Then Cay(B :S n ) ha a b-alernaing Hamilon cycle Q. Furher, Q can be choen o ha for eery i 2 [n], Q ha an (i )- inerion pair, and for each i 2 [n] here i ome 2 [n] for which Q ha wo diinc 8

9 (i )-inerion pair. Proof. If he ordinary combinaion ht B bi i minimal, hen by Lemma2imu be iomorphic o one of he combinaion in Figure 2, 3, or 5, each hown wih a cycle Q aifying he condiion of he heorem. Oherwie, aume induciely ha he heorem i rue for all ordinary combinaion wih fewer erice han T B.SincehT B bi i no minimal, T B conain a leaf,, no inciden wih he edge labeled b, uchhaht B ; bi i an ordinary combinaion. Le z be he unique erex of T B adacen o. The Cayley graph of S n on he e B nf(z)g ha n componen G 1 G 2 ::: G n, where G k i he ubgraph of Cay(B :S n ) induced by all permuaion wih () =k. Le G 0 denoe he Cayley graph of permuaion of [n] nfg, generaed by he e B nf(z)g. Then he inducion hypohei hold for G 0.EachG k i iomorphic o G 0, o by inducion, G, in paricular, ha a b-alernaing Hamilon cycle Q. Furher, for each i aifying i 6=, 6=, Q ha an (i )-inerion pair and for each i 6= here i ome 6= for which Q ha wo (i )-inerion pair. For k 6=, inerchanging k and in eery permuaion on Q gie a b-alernaing Hamilon cycle, Q k,ing k. Now, o obain he deired cycle Q for Cay(B :S n ), each of he cycle Q k, where k 6=, i pliced ino he cycle Q aa(z k)-inerion pair of Q (z i he unique erex of T B adacen o.) Thi i done a follow (ee Figure 6). Le be he (z k) inerion pair of Q. Le 0 0 be he correponding pair on he cycle Q k. Tha i, 0 i obained from by inerchanging and k, and imilarly for and 0. Then imply delee edge and 0 0 and add edge 0 and 0 correponding o he generaor (z) inb. I remain o how haaferallq k are pliced ino Q o form Q, ha here i ill an (i )-inerion pair for eery i 2 [n] and ha for each i, here i ome for which Q ha wo (i )-inerion pair. Fir conider i 6= z and 6=. The cycle Q ha an (i )-inerion pair and for ome 6= here are wo (i )-inerion pair. Thee pair are ill in he nal cycle 9

10 Q Q k = = z # # # # k k u u z k k = = 0 0 Figure 6: Splicing cycle Q k ino cycle Q a a (z k) inerion pair in proof of Theorem 1. = = # # # # # # k k l l Q l l z i z i # # # k k Q k k k Q l z i l l l l Figure 7: Coneraion of (i )-inerion pair when i 6= z and 6=. l l = = = =

11 Q unle ome Q k wa pliced ino Q a a (z k)-inerion pair which wa alo an (i )-inerion pair. Bu hen, for any l 6= k conider he conecuie pair on Q l obained by wapping elemen and l in each of (ee Figure 7). Then i an (i )-inerion pair on Q l. Noe ha (z) = (z) =k. Bu, in plicing Q l ino Q, Q l i pli only a a pair wih elemen in poiion z, o i ill an (i )-inerion pair in Q. Thu, afer plicing, here i no ne lo in inerion pair for i 6= z and 6=. For i 6= z and =, chooe k 6=. InQ here wa an (i k)-inerion pair. Inerchanging elemen and k in each of gie an (i )-inerion pair on Q k. Since i 6= z, hi i no he pair in Q k which wa pli when Q z wa pliced ino Q. Thu each Q k, k 6= conribue an (i )-inerion pair o Q. If i =, he number of ( k)-inerion pair on Q k i (n ; 1)!=2. During he plicing, only one pair i pli for k 6= and only n;1 pair for k =. So,Q conain a( k)-inerion pair for eery k, awell a wo ( )-inerion pair. In he cae ha i = z, plicing Q k ino Q for k 6= can only pli Q k a a (z )-inerion pair for =. So,een afer plicing, Q k will conain (z )-inerion pair for eery 6= k. Chooe any l m, diinc from k and. Then each ofq l and Q m conain a (z k)-inerion pair, een afer plicing. Finally, wemu check fora(z )-inerion pair. The cycle Q ha none and each Q k, k 6= ge pli a a (z )-inerion pair during plicing. inducion, here i ome 6= for which Q Howeer, by conain wo diinc (z )-inerion pair. Correponding o hee, Q conain wo (z )-inerion pair. Thu, een afer plicing, Q conain a (z )-inerion pair. 2 3 Sar and Flare: Excepional Combinaion Le B be a bai of ranpoiion for S n and b 2 B. We conider here he cae where T B i a ar or a are in which b oin he erex of degree 2 wih a leaf (ee Figure 4.) In boh cae, any wo edge in B nfbg are adacen, o he echnique ued for 11

12 b Figure 8: Bai cae for are (read acro.) ordinary combinaion will no work. We focu aenion on are, and hen how ha ar can be handled imilarly. If T B i a are, we can aume ha B = F n = f(12) (23) (34) (35) ::: (3n)g. For n 5 are are iomorphic o ordinary combinaion, unle b = (12). We hownow ha een in hi cae, Cay(F n :S n ) ha a (12)-alernaing Hamilon cycle. Theorem 2 For n 5, Cay(F n :S n ) ha a (12)-alernaing Hamilon cycle H aifying 1. For n odd, here areconecuie permuaion n n on H aifying n (3) = 2 n (n ; 1) = 1 n (n) =n n (3) = n n (n ; 1) = 1 n (n) =2 12

13 2. For 0 k<(n ; 1)=2 when n i een and for 1 k<(n ; 1)=2 when n i odd, here areconecuie permuaion (k) n and (k) n on H aifying (k) n (3) = 2k +1 (k) n (n) =2k +2 (k) n (3) = 2k +2 (k) n (n) =2k +1 Proof. The heorem i rue when n = 5, a demonraed in Figure 8. Noe ha on he cycle of Figure 8, he required conecuie permuaion 5 and 5 are and The required conecuie permuaion (1) 5 and (1) 5 are and (Noe ha he order doe no maer a long a he permuaion appear conecuiely.) Aume ha for ome n 5, Cay(F n :S n ) ha a (12)-alernaing Hamilon cycle H aifying condiion (1) and (2) of he heorem. If we appendn + 1 o eery permuaion on H, wehae ab-alernaing cycle in Cay(F n+1 :S n+1 ), call i H n+1, ill aifying (1) and (2). For 1 i n, he ubgraph of Cay(F n+1 :S n+1 ), induced by he elemen ofs n+1 wih i in poiion n+1, i iomorphic o Cay(F n :S n ), and herefore i conain a (12)- alernaing Hamilon cycle H i. Noe ha gien any permuaion wih (n +1)= i, and any ranpoiion of he form (3k) 2 F n,wemay aume ha i followed by an edge labeled (3k) onh i. (Some edge labeled (3k) mu appear on H i ince F n i a bai for S n. Simply arrange he cyclic li of generaor correponding o he edge along H i o begin wih (3k) and apply hem, aring wih permuaion. Thi yield a new H i wih he required propery.) The idea of he conrucion i o plice H 1 ::: H n ino H n+1 in uch away o obain a (12)-alernaing Hamilon cycle H in Cay(F n+1 :S n+1 ) and preere properie (1) and (2) of he heorem. For n odd, we r plice H 1, H 2,andH n ino H n+1 a he pair 0 n 0 n on H n+1 correponding o n n on H (ee Figure 9.) To do hi we ue he fac ha he following compoiion of ranpoiion i he ideniy: 13

14 A B n 0 = ab 2 1 nn+1 n 0 = ab n 12n+1 cd 2k+1 2k+2 n+1 cd 2k+2 2k+1 n+1 ab 1 n+1 2 n ab n+1 12n cd n+1 2k+1 2k+2 cd 2k+1 n+1 2k+2 ab n n ab 2 n+1 n 1 6 H n+1 H n H 1 H 2 6 new ((n;1)=2) ((n;1)=2) pair n+1 n+1 6 new (0) H 2k+2 H 2k+1 cd n+1 2k+2 2k+1 cd 2k+2 n+1 2k+1 new (k) n+1 (k) n+1 pair ab n+1 1 n 2 ab 1 n+1 n 2 n+1 (0) n+1 pair B A Figure 9: For n odd, plicing he cycle H i ino H n+1. 14

15 (3 n)(3 n+1)(3 n;1)(3 n+1)(3 n)(3 n+1)(3 n;1)(3 n+1) = id We know ha 0 n and 0 n appear conecuiely on H n+1 and, a dicued aboe, we may aume wihou lo of generaliy ha: [ 0 n (3 n+1)] and [ 0 n (3 n+1)](3 n;1) appear conecuiely on H n, [ 0 n(3 n+1)(3 n;1)(3 n+1)] and [ 0 n(3 n+1)(3 n;1)(3 n+1)](3 n) appear conecuiely on H 1, and [n 0 (3 n+1)(3 n;1)(3 n+1)(3 n)(3 n+1)] and [ 0 n (3 n+1)(3 n;1)(3 n+1)(3 n)(3 n+1)](3 n;1) appear conecuiely on H 2. In each pair aboe, a well a for he pair 0 n 0 n, delee he edge oining he wo elemen of he pair in heir repecie cycle. Then ue edge correponding o he generaor (3 n+1) o oin ogeher he cycle a hown in Figure 9. Noe from Figure 9 ha hi conrucion proide u wih he required pair (0) n+1 (0) n+1 and ((n;1)=2) n+1 ((n;1)=2) n+1 for he (12)-alernaing cycle H being conruced in Cay(F n+1 :S n+1 ). For 0 k<(n ; 1)=2 when n i een and 1 k<(n ; 1)=2 when n i odd, we plice H 2k+1 and H 2k+2 ino H n+1 a he conecuie paironh n+1 correponding o (k) n n (k) on H, imilar o he mehod aboe, bu uing he ideniy (3 n)(3 n+1)(3 n)(3 n+1)(3 n)(3 n+1) = id (ee Figure 9 and 10.) Noe from Figure 9 ha hi proide for he cycle H he pair (k) n+1 (k) n+1 for 1 k<(n ; 1)=2 and, from Figure 10, when n i een, gie n+1 n+1. 2 The cae of ar can be handled imilarly. For a bai B of ranpoiion of S n, if T B i a ar, we may aume ha B = R n = f(31) (32) (34) (35) :::(3n)g, and ha he diinguihed edge b of B i (32) (ee Figure 4.) In hi cae we hae he following heorem. 15

16 A H n+1 H 2 H 1 (0) n = ab 1 2 n+1 n (0) = ab 2 1 n+1 ab n ab 1 n+1 2 ab 2 n+1 1 ab n+1 21 A new n+1 n+1 pair B (k) n = cd 2k+22k+1 n+1 cd n+1 2k+1 2k+2 n (k) = cd 2k+12k+2 n+1 cd 2k+1 n+1 2k+2 H 2k+2 H 2k+1 6 cd n+1 2k+2 2k+1 cd 2k+2 n+1 2k+1 new (k) n+1 (k) n+1 pair B Figure 10: For n een, plicing he H i ino H n+1 16

17 Theorem 3 For n 5, Cay(R n :S n ) ha a (32)-alernaing Hamilon cycle H aifying 1. For n odd, here areconecuie permuaion n n on H aifying n (3) = 2 n (n ; 1) = 1 n (n) =n n (3) = n n (n ; 1) = 1 n (n) =2 2. For 0 k<(n ; 1)=2 when n i een and for 1 k<(n ; 1)=2 when n i odd, here areconecuie permuaion (k) n and (k) n on H aifying (k) n (3) = 2k +1 (k) n (n) =2k +2 (k) n (3) = 2k +2 (k) n (n) =2k +1 Proof. The heorem i rue when n = 5, a demonraed in Figure 11. Noe ha on he cycle of Figure 11, he required conecuie permuaion 5 and 5 are and The required conecuie permuaion (1) 5 and (1) 5 are and The remainder of he proof i idenical o he proof of Theorem Final Remark There hae been ome oher paper wrien abou nding Hamilon cycle hrough pecied maching in graph, bu no in connecion wih Cayley graph ([Ha],[Wo]). For example, Haggki [Ha] ha hown ha if d(u) +d() V (G) + 1 for all nonadacen erice u and of G, heng ha a Hamilon pah hrough any gien perfec maching. By deleing all odd permuaion from our li we obain liing of he alernaing group A n. In he cae of a ar, where B = f(1 n) (2 n) ::: (n;1 n)g and b =(1n), noe ha ince (1 n)( n) =(1n), our reul proide anoher proof of he reul of Gould and Roh [GoRo] ha he digraph Cay(X :A n ) i hamilonian for n 5, where X = f(1 n) 1 <<ng. 17

18 b 2 Figure 11: Bai cae for ar (read acro.) Tchuene [Tc] howed ha here i a Hamilon pah beween any wo permuaion of oppoie pariy incay(b :S n )forany bai of ranpoiion B. The nex lemma how ha i i no in general he cae ha here i a b-alernaing pah conaining M b beween any wo permuaion of oppoie pariy. Le ht B bi be a combinaion. If he edge b i remoed from T B hen wo ree remain hee ree induce a pariion of [n] ino wo e, ay X and Y. Lemma 3 Le X Y be he pariion of [n] induced byht B bi. Any b-alernaing Hamilon pah in Cay(B :S n ) ha ar a he permuaion and end a he permu- 18

19 aion 0 mu aify he following condiion. [ (i) = [ 0 (i) i2x i2x Proof. Conider he muligraph M formed from Cay(B :S n )by condening ino a ingle erex, for each k-ube S of [n], hoe permuaion for which f(i) i 2 Xg = S. Thu M ha n k erice and each erex i regular of degree k!(n ; k)!. Eery edge of M i labeled b ince eery ranpoiion oher han b eiher wap wo elemen wih poiion in X or wap wo elemen wih poiion in Y. A b- alernaing pah in in Cay(B :S n ) ha conain eery edge of M b become an Euler our in M. Clearly, hi our ha o ar and end a he ame condened erex. 2 If n = 4 hen here are wo non-iomorphic excepional combinaion ht B bi, namely he ar B = f(12) (13) (14)g wih b = (12), and he pah B = f(12) (23) (34)g again wih b = (12). In hee cae i i no oo dicul o how ha here i no b-alernaing Hamilon cycle. Howeer, here are b-alernaing Hamilon pah conaining M b a hown in Figure 12. Below we li ome queion for furher ineigaion. 1. I here an ecien algorihm o generae he permuaion on a b-alernaing Hamilon cycle? We would like an algorihm whoe oal orage requiremeni O(n) and whoe oal running ime i O(n!). A raighforward implemenaion of our proof lead o algorihm ha require (nn!) ime and (nn!) pace. 2. I he neceary condiion of Lemma 3, ogeher wih he condiion ha and 0 hae oppoie pariy, alo a ucien condiion for he exience of a Hamilon pah from o 0?We conecure ha he condiion i ucien. 3. Gien a maching M in he n-cube Q n, i here a Hamilon cycle in Q n ha include eery edge of M? 19

20 Figure 12: (12)-alernaing Hamilon pah. 4. If X i a e of generaor for a group G, andx 2 X i an inoluion (i.e., x 2 = id), hen x induce a perfec maching, M x,in Cay(X :G). A naural queion i wheher here i a x-alernaing pah in Cay(X :G). In general, here i no x-alernaing Hamilon pah in Cay(X : G). For example, if X = f(1 2) (1 2 n)g, wheren 3 i odd, hen he following lile argumen, imilar o he proof of Lemma 3, how ha Cay(X :S n ) ha no (1 2)- alernaing pah. Condene ino ingle uper-erice all hoe permuaion inequialen under he roaion (1 2 n). The reuling muligraph ha (n;1)! erice, each of degree n,andanyhamilon pah in Cay(X :S n ) become a Eulerian cycle in he muligraph. Clearly, here i no Eulerian cycle if n i odd. On he oher hand, here doe appear o be a x-alernaing pah in Cay(X :G) 20

21 if G i a reecion group and x i one of i generaor, bu we hae no proen hi fully. In [CoSlWi], he Cayley graph of reecion group are hown o be Hamilonian. Reference [CoSlWi] [CuWi] [GoRo] [Ha] [o] [KeWi] [KoLi] [Lo] [PrRu] [RuSa] [Sl]. H. Conway, N.. A. Sloane, and A. R. Wilk,\Gray code for reecion group," Graph and Combinaoric, 5 (1989) , S.. Curran and D. Wie, \Hamilon pah in careian produc of direced cycle," in Cycle in Graph, Annal of Dicree Mahemaic 27, B. R. Alpach and C. D. Godil, ed., Norh Holland, The Neherland (1985). R.. Gould and R. Roh, \Cayley digraph and (1 n) equencing of he alernaing group A n," Dicree Mahemaic 66 (1987) R. Haggki, \On F-Hamilonian Graph," in Graph Theory and Relaed Topic, Academic Pre, New York and London, (1979) S. M. ohnon, \Generaion of permuaion by adacen ranpoiion," Mah. Comp. 17 (1963) K. Keaing and D. Wie, \On Hamilon cycle in Cayley graph of group wih cyclic commuaor ubgroup," in Cycle in Graph, B. R. Alpach and C. D. Godil, ed., Annal of Dicree Mahemaic 27, Norh Holland, The Neherland (1985). V. L. Kompel'makher and V. A. Likoe, \Sequenial generaion of arrangemen by mean of a bai of ranpoiion," Kiberneica, 3 (1975) L. Loaz, Problem 11 in Combinaorial Srucure and heir Applicaion, Gorden and Breach, London G. Pruee and F. Rukey, \Generaing he linear exenion of cerain poe by ranpoiion," SIAM. Dicree Mah., 4 (1991) F. Rukey and C. Saage, \Generaing permuaion by rericed adacen ranpoiion," unpublihed manucrip (1989). P.. Slaer, \Generaing all permuaion by graphical ranpoiion," ARS Combinaoria 5 (1978) [S] H. Seinhau, One Hundred Problem in Elemenary Mahemaic, Baic (1964). [Tc] M. Tchuene, \Generaion of permuaion by graphical exchange," ARS Combinaoria 14 (1982)

22 [Tr] [Wo] H. F. Troer, \PERM (Algorihm 115)", Communicaion of he ACM 5, No. 8 (1962) A.P. Woda, \Hamilonian Cycle Through Maching", Demonraio Mah., 22 (1988)