Research Article On Cayley Digraphs That Do Not Have Hamiltonian Paths

Transcription

1 Hindwi Pulishing Corportion Interntionl Journl of Comintorics Volume 2013, Article ID , 7 pges Reserch Article On Cyley Digrphs Tht Do Not Hve Hmiltonin Pths Dve Witte Morris Deprtment of Mthemtics nd Computer Science, University of Lethridge, Lethridge, AB, Cnd T1K 3M4 Correspondence should e ddressed to Dve Witte Morris; dve.morris@uleth.c Received 23 June 2013; Accepted 4 Novemer 2013 Acdemic Editor: Jun-Ming Xu Copyright 2013 Dve Witte Morris. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. We construct n infinite fmily { Cy(G i ; i ; i )} of connected, 2-generted Cyley digrphs tht do not hve hmiltonin pths, such tht the orders of the genertors i nd i re unounded. We lso prove tht if G is ny finite group with [G, G] 3,thenevery connected Cyley digrph on G hs hmiltonin pth (ut the conclusion does not lwys hold when [G, G] = 4 or 5). 1. Introduction Definition 1. For suset S of finite group G, thecyley digrph Cy(G; S) is the directed grph whose vertices re the elements of G nd with directed edge g gs for every g G nd s S. The corresponding Cyley grph is the underlying undirected grph tht is otined y removing the orienttions from ll the directed edges. It hs een conjectured tht every (nontrivil) connected Cyley grph hs hmiltonin cycle. (See the iliogrphy of [1]forsomeofthelitertureonthisprolem.)Thisconjecture does not extend to the directed cse, ecuse there re mny exmples of connected Cyley digrphs tht do not hve hmiltonin cycles. In fct, infinitely mny Cyley digrphs do not even hve hmiltonin pth. Proposition 2 (ttriuted to Milnor [2, p.201]). Assume the finite group G is generted y two elements nd, suchtht 2 = 3 = e.if G 9 2, then the Cyley digrph Cy(G;,)does not hve hmiltonin pth. The exmples in the ove proposition re very constrined, ecuse the order of one genertor must e exctly 2 nd the order of the other genertor must e exctly 3. In this note, we provide n infinite fmily of exmples in which the orders of the genertors re not restricted in this wy. In fct, nd cnotheofritrrilylrgeorder. Theorem 3. For ny n N,thereisconnectedCyleydigrph Cy(G;, ),suchtht (1) Cy(G;, ) does not hve hmiltonin pth, (2) nd oth hve order greter thn n. Furthermore, if p is ny prime numer such tht p > 3 nd p 3(mod 4), then we my construct the exmple so tht the commuttor sugroup of G hs order p. Moreprecisely,G= Z m Z p is semidirect product of two cyclic groups, so G is metcyclic. Remrk 4. Hereresomereltedopenquestionsndother comments. (1) The ove results show tht connected Cyley digrphs on solvle groups do not lwys hve hmiltonin pths. On the other hnd, it is n open question whether connected Cyley digrphs on nilpotent groups lwys hve hmiltonin pths. (See [3] for recent results on the nilpotent cse.) (2) The ove results lwys produce digrph with n even numer of vertices. Do there exist infinitely mny connected Cyley digrphs of odd order tht do not hve hmiltonin pths? (3) We conjecture tht the ssumption p 3(mod 4) cn e eliminted from the sttement of Theorem 3. On the other hnd, it is necessry to require tht p>3 (see Corollry 16).

2 2 Interntionl Journl of Comintorics (4) If G is elin, then it is esy to show tht every connected Cyley digrph on G hs hmiltonin pth. However, some elin Cyley digrphs do not hve hmiltonin cycle. See Section 5 for more discussion of this. (5) The proof of Theorem 3 ppers in Section 3, fter some preliminries in Section Preliminries We recll some stndrd nottion, terminology, nd sic fcts. Nottion.LetG e group, nd let H e sugroup of G.(All groupsinthispperressumedtoefinite.) (i) e is the identity element of G; (ii) x g =g 1 xg,forx, g G; (iii) we write H Gto sy tht H is norml sugroup of G; (iv) H G = h g h H,g G isthenorml closure of H in G,soH G G. Definition 5. Let S e suset of the group G. (i) H= SS 1 is the rc-forcing sugroup,wheress 1 = {st 1 s,t S}. (ii) For ny S, 1 H is clled the terminl coset.(this is independent of the choice of.) (iii) Any left coset of H tht is not the terminl coset is clled regulr coset. (iv) For g G nd s 1,...,s n S,weuse[g](s i ) n i=1 to denote the wlk in Cy(G; S) tht visits (in order) the vertices g, gs 1,gs 1 s 2,...,gs 1 s 2 s n. (1) We usully omit the prefix [g] when g = e.also, we often use nottion when sequences re to e conctented. For exmple, ( 4,(s i ) 3 i=1,t j) 2 j=1 = (,,,, s 1,s 2,s 3,t 1,,,,,s 1,s 2,s 3,t 2 ). Remrk 6. Here re two oservtions out the rc-forcing sugroup. (1) It is importnt to note tht SS 1 Sg,forevery g G.Furthermore,wehve SS 1 = S 1,for every S. (2) It is sometimes more convenient to define the rcforcing sugroup to e S 1 S, insted of SS 1 (e.g.,thisistheconventionusedin[3, p.42]).the difference is minor, ecuse the two sugroups re conjugte: for ny S, wehve (2) S 1 S = 1 S = S 1 = SS 1. (3) Definition 7. Suppose L is hmiltonin pth in Cyley digrph Cy(G; S) nd s S. (i) A vertex g Gtrvels y s if L contins the directed edge g gs. (ii) A suset X of G trvels y s if every element of X trvels y s. Lemm 8 (see Housmn [4, p.82]). Suppose L is hmiltonin pth in Cy(G;, ), withinitilvertexe, ndleth = 1 e the rc-forcing sugroup. Then, (1) the terminl vertex of L elongs to the terminl coset 1 H, (2) ech regulr coset either trvels y or trvels y. 3. Proof of Theorem 3 Let (i) α e n even numer tht is reltively prime to (p 1)/2,withα>n; (ii) β multipleof(p 1)/2 tht is reltively prime to α, with β>n; (iii) genertor of Z α ; (iv) genertor of Z β ; (v) z genertor of Z p ; (vi) r primitiverootmodulop; (vii) G=(Z α Z β ) Z p,wherez =z 1 nd z =z r2 ; (viii) =z,so = α,ndinverts Z p ; (ix) = z, so = β, nd cts on Z p vi n utomorphism of order (p 1)/2; (x) H= 1 = 1 =Z α Z β. Suppose L is hmiltonin pth in Cy(G;, ). This will led to contrdiction. Itiswellknown(ndesytosee)thtCyleydigrphs re vertex-trnsitive, so there is no hrm in ssuming tht the initil vertex of L is e.notetht (i) the terminl coset is 1 H=z 1 H; (ii) since p 3(mod 4),wehveZ p = 1,r2. Cse 1. Assume t most one regulr coset trvels y in L. Choose z Z p,suchthtz H is regulr coset, nd ssume it is the coset tht trvels y, if such exists. For g G,let Letting p =(p 1)/2,wehve B g = {g k H k Z}. (4) (r 2 ) p 1 +(r 2 ) p 2 + +(r 2 ) 1 +1 = (r2 ) p 1 r 2 = rp r 2 0 (mod p), 1 (5)

3 Interntionl Journl of Comintorics 3 so (p 1)/2 =(z) p = p z (r2 ) p 1+(r 2 ) p 2 + +(r 2 ) 1 +1 = p Z α Z β =H. Therefore #B e (p 1)/2 p 2,sowecnchoosetwo cosets z i H nd z j H tht do not elong to B e. Recll tht, y definition, z H is not the terminl coset z 1 H,soz z is nontrivil element of Z p.then,sincez p = 1, r 2, we cn choose some h, = H, suchtht (z j i ) h =z z.now,since z i H, z j H B e, z 1 h 1 z j i z 1 (z j i ) h H=z 1 (z z)h=z H, we my multiply on the left y g=z 1 h 1 z i to see tht (6) (7) z 1 H, z H B g. (8) Therefore, no element of B g is either the terminl coset or the regulr coset tht trvels y. This mens tht every coset in B g trvels y, sol contins the cycle [g]( β ),which contrdicts the fct tht L is (hmiltonin) pth. Cse 2. Assume t lest two regulr cosets trvel y in L.Let z i H nd z j H e two regulr cosets tht oth trvel y.since Z p = 1,r2, we cn choose some h, = H,suchtht (z 1 ) h =z j i. Note tht z i h 1 k trvels y,foreveryk Z. (i) If k=2lis even, then k = (z) 2l = (zz) l =( 2 z z) l =( 2 z 1 z) l = 2l H, so z i h 1 k z i H trvels y. (ii) If k=2l+1is odd, then (9) k = (z) 2l+1 = (z) 2l (z) = 2l (z) = k z, (10) so z i h 1 k =z i h 1 ( k z) = z i h 1 z 1 k trvels y. =z i (z 1 ) h h 1 k z i (z j i )H=z j H (11) Therefore L contins the cycle [z i h 1 ]( α ),whichcontrdicts the fct tht L is (hmiltonin) pth. 4. Cyclic Commuttor Sugroups of Very Smll Order It is known tht if [G, G] = 2, theneveryconnectedcyley digrph on G hs hmiltonin pth. (Nmely, we hve [G, G] Z(G), so G is nilpotent, nd the conclusion, therefore, follows from Theorem 14(2) elow.) In this section, we prove the sme conclusion when [G, G] = 3.Welsoprovide counterexmples to show tht the conclusion is not lwys true when [G, G] = 4 or [G, G] = 5. We egin with severl lemms. The first three ech provide wy to convert hmiltonin pth in Cyley digrph on n pproprite sugroup of G to hmiltonin pth in Cyley digrph on ll of G. Lemm 9. Assume (i) G is finite group, such tht [G, G] Z p k,wherep is prime nd k N; (ii) S is generting set for G; (iii), S,suchtht [, ] = [G, G]; (iv) N=,. If Cy(N;, ) hs hmiltonin pth, then Cy(G; S) hs hmiltonin pth. Proof. Since [G, G] N, weknowthtg/n is n elin group, so there is hmiltonin pth (s i ) m i=1 in Cy(G/N; S) (see Proposition 19 elow). Also, y ssumption, there is hmiltonin pth (t j ) n j=1 in Cy(N;,).Then is hmiltonin pth in Cy(G; S). (((t j ) n j=1,s i) m i=1,(t j) n j=1 ) (12) Definition 10. If K is sugroup of G, thenk\ Cy(G; S) denotes the digrph whose vertices re the right cosets of K in G nd with directed edge Kg Kgs for ech g Gnd s S.NotethtK\ Cy(G; S) = Cy(G/K; S) if K G. Lemm 11 ( Skewed-Genertors Argument, cf. [3, Lem. 2.6], [5,Lem.5.1]). Assume (i) S is generting set for the group G; (ii) K is sugroup of G,suchthteveryconnectedCyley digrph on K hs hmiltonin pth; (iii) (s i ) n i=1 is hmiltonin cycle in K\ Cy(G; S); (iv) Ss 2 s 3 s n =K. Then Cy(G; S) hs hmiltonin pth.

4 4 Interntionl Journl of Comintorics Proof. Since Ss 2 s 3 s n =K,weknowthtCy(K; Ss 2 s 3 s n ) is connected, so, y ssumption, it hs hmiltonin pth (t j s 2 s 3 s n ) m j=1.then ((t j,(s i ) n i=2 )m 1 j=1,t m,(s i ) n 1 i=2 ) (13) is hmiltonin pth in Cy(G; S). Lemm 12. Assume (i) S isgenerting set ofg, with rc-forcing sugroup H = SS 1 ; (ii) there is hmiltonin pth in every connected Cyley digrph on H G ; (iii) either H=H G,orHiscontined in unique mximl sugroup of H G. Then Cy(G; S) hs hmiltonin pth. Proof. It suffices to show tht there exists hmiltonin cycle (s i ) n i=1 in such tht H G = Ss 2 s n, Cy ( G ;S), HG ( ) for then Lemm 11 provides the desired hmiltonin pth in Cy(G; S). If H G =H,theneveryhmiltonincycleinCy(G/H G ; S) stisfies ( ) (see Remrk 6 (1)). Thus, we my ssume H G =H, so, y ssumption, H is contined in unique mximl sugroup M of H G.SinceH G is generted y conjugtes of S 1 S (see Remrk 6 (2)), there exist,, c S, such tht ( 1 ) c M. We my lso ssume H G =G(since, y ssumption, every Cyley digrph on H G hs hmiltonin pth), so, letting n= G : H G 2,wehvethetwohmiltonincycles( n 1,c)nd ( n 2,,c)in Cy(G/H G ;S).Since ( n 1 c) 1 ( n 2 c) = ( 1 ) c M, (14) the two products n 1 c nd n 2 c cnnot oth elong to M. Hence, either ( n 1,c) or ( n 2,,c) is hmiltonin cycle (s i ) n i=1 in Cy(G/H G ;S),suchthts 1 s 2 s n M.SinceM istheuniquemximlsugroupofh G tht contins H,this implies s desired. H G = H,s 1 s 2 s n = Ss 2 s 3 s n, (15) The finl hypothesis of the preceding lemm is utomticlly stisfied when [G, G] is cyclic of prime-power order. Lemm 13. If [G, G] is cyclic of order p k,wherep is prime, nd H is ny sugroup of G, then either H=H G or H is contined in unique mximl sugroup of H G. Proof. Note tht the norml closure H G is the (unique) smllest norml sugroup of G tht contins H. Therefore, H G H[G,G](since H [G, G] is norml in G). This implies tht if M is ny proper sugroup of H G tht contins H,then M=H (M [G, G]) H (H G [G, G]) p. (16) Therefore, H (H G [G, G]) p is the unique mximl sugroup of H G tht contins M. The following known result hndles the cse where G is nilpotent. Theorem 14 (see Morris [3]). Assume G is nilpotent, nd S genertes G.Ifeither (1) #S 2or (2) [G, G] = p k,wherep is prime nd k N, then Cy(G; S) hs hmiltonin pth. We now stte the min result of this section. Theorem 15. Suppose (i) [G, G] is cyclic of prime-power order, (ii) every element of G either centrlizes [G, G] or inverts it. Then every connected Cyley digrph on G hs hmiltonin pth. Proof. Let S e generting set for G.Write[G, G] = Z p k for some p nd k. Since every miniml generting set of Z p k hs only one element, there exist, S, such tht [, ] = [G, G].Then,yLemm 9, we my ssume S = {, }. Let H = 1 e the rc-forcing sugroup. We my ssume H G =G,forotherwisewecouldssume,yinduction on G, tht every connected Cyley digrph on H G hs hmiltonin pth, nd then Lemm 12 would pply (since Lemm 13 verifies the remining hypothesis). So HZ p k =H[G, G] H G =G. (17) If nd oth invert Z p k,thenh = 1 centrlizes Z p k = [G, G],soGis nilpotent. Then Theorem 14 pplies. Therefore, we my now ssume tht does not invert Z p k. Then, y ssumption, centrlizes Z p k.letn= G:H,nd write =z, where Hnd z Z p k.then=z Hz nd = ( 1 )(z) Hz. Since, = G, thisimplies H z = G. Therefore, [H] ( n )=[H,Hz,Hz 2,...,Hz n 1,H] (18) is hmiltonin cycle in H\ Cy(G; S),soLemm 11 pplies. Corollry 16. If [G, G] 3 or [G, G] Z 4,thenevery connected Cyley digrph on G hs hmiltonin pth.

5 Interntionl Journl of Comintorics 5 Proof. Theorem 15 pplies, ecuse the groups {e} nd Z 2 hve no nontrivil utomorphisms, nd inversion is the only nontrivil utomorphism of Z 3 or Z 4. Remrk 17 ([6, p. 266]). In the sttement of Corollry 16,the ssumption tht [G, G] Z 4 cnnot e replced with the weker ssumption tht [G, G] = 4. For counterexmple, let G = A 4 Z 2.Then [G, G] = 4, utitcneshown withoutmuchdifficultythtcy(g;,) does not hve hmiltonin pth when = ((1 2)(3 4), 1) nd = ((1 2 3), 0). Here is counterexmple when [G, G] = 5. Exmple 18. Let G=Z 12 Z 5 = h z, wherez h =z 3. Then [G, G] = 5, nd the Cyley digrph Cy(G; h 2 z, h 3 z) is connected ut does not hve hmiltonin pth. Proof. A computer serch cn confirm the nonexistence of hmiltonin pth very quickly, ut, for completeness, we provide humn-redle proof. Let =h 2 z=z 4 h 2 nd =h 3 z=z 3 h 3.Thergument in Cse 2 of the proof of Theorem 3 shows tht no more thn one regulr coset trvels y in ny hmiltonin pth. On the other hnd, since hmiltonin pth cnnot contin ny cycle of the form [g]( 4 ),weknowthttlest ( G 1)/4 = 14 vertices must trvel y.since 1 =12<14,thisimplies tht some regulr coset trvels y. So exctly one regulr coset trvels y in ny hmiltonin pth. For 0 i 3nd 0 m 11,letL i,m e the spnning sudigrph of Cy(G;, ) in which (i) ll vertices hve outvlence 1, except 1 ( 1 ) m = z 4 h 9 m, which hs outvlence 0; (ii)theverticesintheregulrcosetz i H trvel y ; (iii) vertex 1 h j =z 4 h 9 j in the terminl coset trvels y if 0 j<m; (iv) ll other vertices trvel y. An oservtion of D. Housmn [7, Lem. 6.4()] tells us tht if L is hmiltonin pth from e to 1 ( 1 ) m,inwhichz i H is theregulrcosetthttrvelsy,thenl=l i,m.thus,from the conclusion of the preceding prgrph, we see tht every hmiltonin pth (with initil vertex e)musteequltol i,m, for some i nd m. However, L i,m is not (hmiltonin) pth. More precisely, for ech possile vlue of i nd m, the following list displys cyclethtiscontinedinl i,m : (i) if i=0nd 0 m 8, z 2 h 3 zh 6 z 3 h 9 z 4 z 2 h 3 ; (19) (ii) if i=0nd 9 m 11, h 2 zh 4 h 3 z 4 h 7 z 2 h 5 zh 9 z 3 h 8 (iii) if i=1nd 0 m 7, h 4 z 3 h 7 z 2 h 10 (iv) if i=1nd 8 m 11, h zh 4 z 3 h 5 h 6 z 4 h 8 (v) if i=2nd 0 m 9, h 5 zh 8 z 2 h 9 z 4 h 11 (vi) if i=2nd 10 m 11, z 2 h 3 z 4 h 5 z 2 h 11 (vii) if i=3nd 0 m 10: h 7 z 4 h 10 (viii) if i=3nd m=11, z 3 h 2 z 4 h 4 z 3 h 10 z 2 zh 11 z 4 h z 3 z 3 h 10 z 3 h 2 z 2 h 7 z 4 h zh h 2 ; (20) h 4 ; (21) zh 3 h; (22) h 5 ; (23) z 4 h 9 z 2 h 3 ; z 2 h 4 z 3 h 6 z 4 (24) h 7 ; (25) z 4 h 8 z 3 h 2. (26) Since L i,m is never hmiltonin pth, we conclude tht Cy(G;, ) does not hve hmiltonin pth. 5. Nonhmiltonin Cyley Digrphs on Aelin Groups When G is elin, it is esy to find hmiltonin pth in Cy(G; S). Proposition 19 (see [6, Thm. 3.1]). Every connected Cyley digrph on ny elin group hs hmiltonin pth. On the other hnd, it follows from Lemm 8(2) tht sometimes there is no hmiltonin cycle. Proposition 20 (see Rnkin [8, Thm. 4]). Assume G=, is elin. Then there is hmiltonin cycle in Cy(G;, ) if nd only if there exist k, l 0,suchtht k l = 1,nd k + l = G : 1.

6 6 Interntionl Journl of Comintorics Exmple 21. If gcd(, n) > 1 nd gcd( + 1, n) > 1, then Cy(Z n ;,+1)does not hve hmiltonin cycle. Allofthenon-hmiltoninCyleydigrphsprovidedy Proposition 20 re 2-generted. However, few 3-generted exmples re lso known. Specificlly, the following result lists (up to isomorphism) the only known exmples of connected, non-hmiltonin Cyley digrphs Cy(G; S), such tht #S >2(nd e S). Theorem 22 (see Locke nd Witte [9]). The following Cyley digrphs do not hve hmiltonin cycles: (1) Cy(Z 12k ;6k,6k+2,6k+3),fornyk Z + ; (2) Cy(Z 2k ;,,+k),for,, k Z +,suchthtcertin technicl conditions (Remrk 23)re stisfied. Remrk 23. The precise conditions in (2) re (i) either or k is odd, (ii) either is even or nd k re oth even, (iii) gcd(, k) = 1,(iv)gcd(, 2k) =1,nd(v)gcd(, k) =1. It is interesting to note tht, in the exmples provided y Theorem 22, thegroupg is cyclic (either Z 12k or Z 2k ), nd either (1) one of the genertors hs order 2 or (2) two of the genertors differ y n element of order 2. S. J. Currn (personl communiction) sked whether the constructions could e generlized y llowing G to e n elin group tht is not cyclic. We provide negtive nswer for cse (2). Proposition 24. Let G e n elin group (written dditively), nd let,, k G,suchthtk is n element of order 2. (Also ssume {,, + k} consists of three distinct, nontrivil elements of G.) If the Cyley digrph Cy(G;,, + k) is connected, ut does not hve hmiltonin cycle, then G is cyclic. Proof. We prove the contrpositive: ssume G is not cyclic, nd we will show tht the Cyley digrph hs hmiltonin cycle (if it is connected). The rgument is modifiction of the proof of [9, Thm. 4.1( )]. Construct sudigrph H 0 of G s in [9, Defn.4.2],ut with G in the plce of Z 2k,with G in the plce of 2k, nd with in the plce of d. (Cse1 is when k ;Cse2 is when k.) Every vertex of H 0 hs oth invlence 1 nd outvlence 1. The rgument in cse 3 of the proof of [9, Thm. 4.1( )] shows tht the Cyley digrph Cy(G;,, + k) hs hmiltonin cycle if, k =G. Therefore, we my ssume, k = G. On the other hnd, we know =G (ecuse G is not cyclic). Since k = 2, this implies G = k. SinceG is not cyclic, this implies tht hs even order. Also, we my write = +k nd = +k for some (unique), nd k,k k.(since,itisesytoseethtk =k,utwedonot need this fct.) Clim. H 0 hs n odd numer of connected components. Arguing s in the proof of [9, Lem. 4.1] (except tht, s efore, Cse 1 is when k nd Cse 2 is when k ), we see tht the numer of connected components in H 0 is G :, k + G :, k if k, (27) G :, k if k. Since =, we know tht one of nd is n even multiple of, nd the other is n odd multiple. (Otherwise, the difference would e n even multiple of, so it would not generte.) Thus, one of G :, k nd G :, k is even, nd the other is odd. So G :, k + G :, k is odd. This estlishes the clim if k. We my now ssume k. This implies tht the element hs odd order (nd k must e nontrivil, ut we do not need this fct). This mens tht is n even multiple of,so must e n odd multiple of (since = ). Therefore, : is odd, which mens G :, k is odd. This completes the proof of the clim. Now, if G :, k is odd, we cn pply very slight modifiction of the rgument in cse 4 of the proof of [9, Thm. 4.1( )]. (Sucse 4.1 is when k nd sucse 4.2 is when k.) We conclude tht Cy(G;,,+k)hs hmiltonin cycle, s desired. Finlly, if G :, k is even, then more sustntil modifictions to the rgument in [9] re required. For convenience, let m = G :, k.notetht,since G :, k is even, the proof of the clim shows tht m is odd nd k. Define H 0 s in sucse 4.1 of [9, Thm. 4.1( )] (with G in the plce of Z 2k nd replcing gcd(, k) with G :, k ). Let H 1 =H 0, nd inductively construct, for 1 i (m + 1)/2,n element H i of E,suchtht {V z V =0,0 y V 2i 2} (28) {V z V =1,x V =0or 1 (mod G :, k )} is component of H i, nd ll other components re components of H 0.TheconstructionofH i from H i 1 isthesmes in sucse 4.1, ut with 2i replced y 2i 1. We now let K 1 =H (m+1)/2 nd inductively construct, for 1 i G :, k /2,nelementK i of E,suchtht {V z V =0} {V z V =1,x V 0,1,..., or 2i 1 (mod G :, k )} (29) is single component of K i.nmely,[9, Lem. 4.2] implies thereisnelementk i =K i 1,suchtht(2i 2), (2i 2)+k, nd (2i 1)+k rellinthesmecomponentofk i.then,for i = G :, k /2,weseethtK i is hmiltonin cycle. Acknowledgments The uthor thnks Stephen J. Currn for sking the question tht inspired Proposition 24. The other results in this pper

7 Interntionl Journl of Comintorics 7 were otined during visit to the School of Mthemtics nd Sttistics t the University of Western Austrli (prtilly supported y funds from Austrlin Reserch Council Federtion Fellowship FF ). The uthor is grteful to collegues there for mking the visit so productive nd enjoyle. References [1]K.Kutnr,D.Mrušič, D. W. Morris, J. Morris, nd P. Šprl, Hmiltonin cycles in Cyley grphs whose order hs few prime fctors, Ars Mthemtic Contemporne, vol. 5, no. 1, pp.27 71,2012. [2] M. B. Nthnson, Prtil products in finite groups, Discrete Mthemtics,vol.15,no.2,pp ,1976. [3] D. W. Morris, 2-generted Cyley digrphs on nilpotent groups hve Hmiltonin pths, Contriutions to Discrete Mthemtics,vol.7,no.1,pp.41 47,2012. [4] D. Housmn, Enumertion of Hmiltonin pths in Cyley digrms, Aequtiones Mthemtice, vol.23,no.1,pp.80 97, [5] D. Witte, Cyley digrphs of prime-power order re Hmiltonin, Journl of Comintoril Theory. Series B, vol.40,no.1, pp , [6] W. Holsztyński nd R. F. E. Strue, Pths nd circuits in finite groups, Discrete Mthemtics, vol. 22, no. 3, pp , [7] S. J. Currn nd D. Witte, Hmilton pths in Crtesin products of directed cycles, in Cycles in Grphs (Burny, B.C., 1982), B.R.AlspchndC.D.Godsil,Eds.,vol.115ofNorth- Hollnd Mth. Stud., pp , North-Hollnd, Amsterdm, The Netherlnds, [8] R. A. Rnkin, A cmpnologicl prolem in group theory, Mthemticl Proceedings of the Cmridge Philosophicl Society,vol.44,pp.17 25,1948. [9]S.C.LockendD.Witte, Onnon-Hmiltonincirculnt digrphs of outdegree three, Journl of Grph Theory, vol.30, no. 4, pp , 1999.

8 Advnces in Opertions Reserch Hindwi Pulishing Corportion Advnces in Decision Sciences Hindwi Pulishing Corportion Journl of Applied Mthemtics Alger Hindwi Pulishing Corportion Hindwi Pulishing Corportion Journl of Proility nd Sttistics The Scientific World Journl Hindwi Pulishing Corportion Hindwi Pulishing Corportion Interntionl Journl of Differentil Equtions Hindwi Pulishing Corportion Sumit your mnuscripts t Interntionl Journl of Advnces in Comintorics Hindwi Pulishing Corportion Mthemticl Physics Hindwi Pulishing Corportion Journl of Complex Anlysis Hindwi Pulishing Corportion Interntionl Journl of Mthemtics nd Mthemticl Sciences Mthemticl Prolems in Engineering Journl of Mthemtics Hindwi Pulishing Corportion Hindwi Pulishing Corportion Hindwi Pulishing Corportion Discrete Mthemtics Journl of Hindwi Pulishing Corportion Discrete Dynmics in Nture nd Society Journl of Function Spces Hindwi Pulishing Corportion Astrct nd Applied Anlysis Hindwi Pulishing Corportion Hindwi Pulishing Corportion Interntionl Journl of Journl of Stochstic Anlysis Optimiztion Hindwi Pulishing Corportion Hindwi Pulishing Corportion