Abela Homegeeous Factorsatos of Grahs Huhog Wu * Shuqu Qa School of Sceces Ashu Uversty Ashu Guzhou 56000 Cha. * Corresog author. Tel.:+86 590853059; emal: hhwu98004@sa.com Mauscrt submtte Jauary 0 06; accete Jue 8 06. o: 0.7706/jam.06.6.3.-9 Abstract: A homogeeous factorsato of a grah Г s a artto of ts arc set such that there exst vertex trastve subgrous M<G Aut(Г) such that M fxes each art setwse of the artto a Greserves the artto a trastve ermutg the arts. I the reset aer we stuy homogeeous factorsato wth M abela. We gve some terestg characterzatos a costructos of such factorsatos. Key wors: Cayley grah homogeeous factorsatos abela grou.. Itroucto Г be a grah wth vertex set VГ a arc set AГ. If there exst a artto { P P P k } of the arc set AГ a two subgrous M<G of Aut such that ) M s trastve o the vertex set V a fxes each P setwse; ) P s a G-varat artto a the uce acto GP of G o P s trastve we call that (ГP) s a (MG)-homogeeous factorsato of ex k. If M s regular o VГ a M G the we call the factorsato a M-Cayley homogeeous factorsato; f artcular M s a abela grou we occasoally smly call that Г has a homogeeous factorsato M-Cayley homogeeous factorsato M-abela homogeeous factorsato a M-crculat homogeeous factorsato. The urose of ths aer s to character M-abela homogeeous factorsato. The fte grahs homogeeous factorsato s tate a researche by famous Algebra grahs theorem exerts Praeger Guralck a Saxl []. I 003 Lm a Strger gave characterzatos for homogeeous factorsatos of comlete grah a Ege-trastve homogeeous factorsatos of comlete grahs [] [3]. I 004 Cuaresma stue homogeeous factorsatos of Johso grah [4]. I 007 Guc L Potock a Praeger accomlshe homogeeous factorsatos of comlete multartte grahs [5]. Fag L a Wag characterze trastve -factorzatos of arc-trastve grahs [6]. The geeral theory of homogeeous factorsato was stue [] [7]. A ecessary a suffcet coto for comlete grahs havg crculat homogeeous factorsatos has bee gve by Praeger a L for comlete grahs havg (MG) crculat homogeeous factorsatos uer the coto that G/M s a cyclc grou [8]. Ths aer gve a research the base of the ma work above.. Costructos R be a grou a M R : H G M : L wth L a subgrou of Aut( M ). Cay( R S) a Cay( R S) for k are Cayley grahs of M. Fally let P A { P P P k } a { S S S k }. Volume 6 Number 3 July 06
Lemma.. Usg otato efe above ( ) s a ( MG ) homogeeous factorsato f a oly f the followg cotos hol: ) H fxes each S. ) S S a setwse S Sj for all j. k 3) s a L varat artto a the uce acto L of L o s trastve. I artcular f we choose M a 3 hol. R the ( ) s a M-Cayley homogeeous factorsato f a oly f the above cotos The followg costructo of M-Cayley homogeeous factorsato s gve [5]. Costructo.. M be a grou a let S be a subset of M \{} reserve by some subgrou H Aut ( M). Cay( M S) O { O O O r } be the set of H-orbts S. For each { r} choose x O. Suose that H has a roer subgrou R cotag R. For each { r} the R-orbt of the elemet let B O be x a let S B B Br. Choose a set { h h h k } of coset reresetatos of R H such that h. Defe S S let P be the arc set of the Cayley grah Cay( M S) { P P P k } a G M : H. The ( ) s a M-Cayey homogeeous factorsato. Lemma.3. R be a grou a Cay( R S) be a Cayley grah. H L be subgrous of Aut ( R S ). Suose L s trastve o S a H s trastve o S say { S S S k } s the set of orbts of H o S. The there exsts ( ) to be a ( MG ) -homogeeous factorsato of ex k where R M R : H a G R : L. Proof: Because L s trastve o S H s trastve o S a H L. We have h Therefore...... H H H S k S S S k. H S ( k) s a block for L. { S l l L} the s a L -varat artto. As S l L we have S ( ) ( ) S for some j. Further L reserves a L l H l lh l H H j j s trastve o { S S S k }. Cay( R S). We coclue ( MG ) satsfes cotos ()-() of Lemma.. So we be sure that has a factorsato whose factors are all of the same valecy. Cay( R S) ( k) P { A A A k } the ( ) s a ( Rˆ : H Rˆ : L ) homogeeous factorsato of ex k. M be a grou wth orer bgger tha. For a oetty automorhsm of M the followg costructo gves a way to costruct a M-Cayley cyclc ( MG ) homogeeous factorsato ( ) G M :. Costructo.4. M a are as above. Suose o( ). The we may choose l such that S : { x M : l x } has at least elemets. That s S s the uo set of all orbts o M say wth whose legth are multle of l. Choose x O a suose t O l O O Ok. Defe l ( t ) l S k{ x x x } a let Sj j S for j 3 l P be the arc set of Cay( M S) { P P Pl } a G M :. Lemma.5. ( ) s a M-Cayley cyclc ( MG ) homogeeous factorsato.. Further let Cay( M S) a for l 3 Volume 6 Number 3 July 06
Proof: let Cay( M S) Aut ( M) o( ). Clam : x S we have x S. Proof: x S. because so a ( x ) x x therefore x S. Accorg to Clam we kow S s the uo set of some -orbts M a Aut ( M S) suose t O l Defe ( t ) l l S k{ x x x } a Sj j S for j 3 l. Suose { S S S l } obvously we kow that reserve a trastve o. Cay( M S) { P P P l } a G M : that s GM we say that ( ) s a M-Cayley cyclc ( MG ) homogeeous factorsato. By costructo.4 we have the followg terestg roosto. Proosto.6. For a gve grou M there s a ( MG ) homogeeous factorsato ( ) for some G f a oly f M. Proof: The suffcecy of the roosto follows rectly by Costructo.4. Sce M s trastve o V t follows M. Further f M the K obvously has o homogeeous factorsato whch roves the ecessty of the roosto. 3. Abela Homogeeous Factorsato of Comlete Grahs F q be a fel of orer q wth q a rme. ( ) GL be the grou of all vertble trasformato a vector sace of meso over fel the set of o-zero vectors of -meso vector sace over fel F q a let V F \{0}. The V ca be vewe F q. It s kow that GL( k ) has atural acto o V a cotag a cyclc subgrou say { } whch s regular o V. we call the cyclc grou { } a Sger subgrou of GL( ). The { } GL( q) GL( ). Usg the Sger subgrou we may rove that comlete grahs of rme ower vertces have abela homogeeous factorsatos of certa ex. Lemma 3.. K be the comlete grah. The Aut( ) S. For each k ( ) a k there exsts a ( MG ) homogeeous factorsato ( ) of ex k wth M cotag a regular cyclc subgrou o V. Proof: R Z be the atve grou of fte fel F. The Cay( R S) wth S R\{0} a Aut ( M) GL( ). Suose { } s a Sger subgrou of GL( ). Sce s cyclc a k ( ) { } Z lk we may choose L { } such that kl a choose H { }. M R : L a G R : H the M G Aut( ) a H : L k. Sce { } s regular o S t follows L a H are sem-regular o S a H has exactly t orbts say O O Ot where t : ( ) H. Choose x O a let be a set of coset reresetatves of L H. Defe S ( x ) L a S S j for X { h h h k } j k. P be the arc set of Cay( M S) { P P P k }. The ( K ) s a ( MG ) homogeeous factorsato of the comlete grah I Lemma 3. f L the M factorsato of the comlete grah t K wth M cotag a regular subgrou R o V. R s regular o V ( K ). That s ( K ) s a R-abela homogeeous K. However the ext lemma roves that the comlete grah K has j h 4 Volume 6 Number 3 July 06
a crculat homogeeous factorsato f a oly f s o. Lemma 3.. K be the comlete grah wth a rme a let M be a cyclc grou of orer of orer actg regular o V. The there exst a artto of A a G N ( ) ( M ) such that ( ) s a M-crculat homogeeous factorsato of ex k f a oly f k ( ). Proof: We frst rove the ecessty of the coto the lemma. Suose G M : H for some H Aut( ). Sce Val( ) t follows k. Suose { P P P k } Cay( M S) wth P A a G M : H for some H. Sce the uce acto G of G s trastve o H s trastve Aut o { S S S k } so k ves the orer of H thus k ( ) as ( ). Note k t easly follows that k. We ow rove the suffcecy of the coto the lemma. Frst as k s o. Ietfyg M wth the atve grou of the rg the Aut( M ) r acts o M by multlcato ( ) where r s a rmtve root of. That s ( ) s the mmal ostve soluto of the equato x r (mo ). S M \{0} a let H Aut ( M). The H H H H has exactly orbts: ( ) a ther legth equal ( ) ( ) resectvely. X { } be a set of the orbts reresetatves of H actg o S by multlcato. Obvously the stablzer H H j for each j so H x : x X H r s a subgrou of H wth ex. Further sce H s abela a k there exsts a subgrou R cotag Hx : x X such that H : R k Costructo. K has a crculat homogeeous factorsato of ex k. I lemma 3.. We rove that comlete grah of rme ower vertces has abela homogeeous factorsato. Ths leas to the followg questo. Questo 3.3. For ot a rme ower whether the comlete grah K has abela homogeeous factorsato? The followg roosto gve a suffcet coto for the exstece of abela homogeeous factorsato of the comlete grahs wth ot rme ower vertces thus roves a artal aswer of Questo 3.3. Proosto 3.4. K r r. By be the comlete grah a let be the rme ower r r factersato of. If ( ) the there exsts a ( MG ) abela homogeeous factorsato ( K ). r r Proof: Suose k ( ) wth k. Wrte l k. M M = Z for r a let M M M Mr. The Aut( M ) GL( ) GL( ) GL( r r). be the Sger subgrou of GL( ) for each. The H : ( ) s a subgrou of Aut( M ) wth l l lr r l l l lr r orer k. Wrte ( ). Sce s sem-regular o M \{} t follows H s sem-regular o M \{} so S : { xm x o( )} M \{} thus by Costructo.4 Cay( M S) K Γ has a M-abela homogeeous factorsato of ex o( ) k. k 5 Volume 6 Number 3 July 06
4. Abela homogeeous factorzato of some comlete multartte grahs Lemma 4.. If M s a grou a L M a subgrou of ex s a orer t the Cay( M M \ L) s somorhc to the comlete multartte grah K st []. Coversely f K st [] s a comlete multartte grah a M a regular grou of automorhsms of the there exsts a subgrou L of orer t a ex s M such that s somorhc to Cay( M M \ L ). roof: See for examle[9 P roosto.] I ths secto we let M P Q P P a Q q ( qare rmes) the Aut ( M) Aut ( P) Aut ( Q) GL( ) GL( q). I GL( ) the sger cyclc grou s GL( ) Z GL( ) so there exsts GL( ) such that o( ) further GL( ) s regular o Z \{}. The same reaso GL( q ) there exsts GL( q) of orer o( ) q q q q. GL( q ) s semregular o Z q \{}. Take ( x y) Aut ( M) where x y such that o( ) o( x) o( y). Costructo 4.. o( ) o( ) q ( q ) Clam: roof: () K q or () K q q. ( x y) x y o( x) o( y) o( ) ( q ) because acts o Z sem-regular let (( a a)()) M ( a a) () q a t s a etty so we f fxe ots of acts o M. That s f fxe ots of x acts o. b x T b F GL( ) 0 x b a a we have ( a a) ( a a) ( a ba a) 0 ( ) ( ) ( a a) ( a a ba ) the a a ba a b s arbtrary so a 0 a (0 a ) s the fxe-ots of x acts o L base o relato of coclue K q. [ ] ( a ) a The same reaso whe o( ) o( ) q or. That s there s fxe-ots. These ots form a cycle grou L M : L q S M \ L q ( q ) by lemma 4. we q ; ( q ) ( q ) the K q [ ]. Obvously q let GL( q) u q : q. ( x y) x y o( ) o( x) o( y) b x b F GL( ) 0 Suose ( a a) Z 0 0 y q 0 Fq \{0} q 0 0 We have 6 Volume 6 Number 3 July 06
x b ( a a) ( a a) ( a ba a) 0 ( ) ( a a) ( a a ba ) the a a ba a b s arbtrary. So a 0 a (0 a ) s the fxe-ots of x acts o ( a ). That s there s fxe-ots. We have ( b b) q 0 q ( b b y ) ( b b ) q ( ) b b ( ) 0 q ( b b ) ( b b ) fxe-ots of y acts o. The q b b q ( b q) a q s arbtrary. So b 0 a ( b 0) s the. That s there s q fxe-ots. Therefore there are q fxe-ots whe acts o q. a b q The ( a0)( b0) q Obvously ths s a abelo grou eote by L a L q M : L q S M \ L q q q( q ). By lemma 4. we ca coclue K q [ q ]. The same reaso we ca raw the followg cocluso (): o( ) o( ) q ( q ) ( q ) the K ; q[ q] (): o( ) o( ) q ( q) the K or K q[ q] q [ q ] Theorem 4.3. M P Q q a q are rme a s ostve teger the Aut ( M) GL( ) GL( q). Suose GL( ) we have The same reaso whe GL( q) the 3 o( ) C C. 3 o( ) q q q Cq Cq q Accorg to the metho of costructo 4. we ca costruct such that Γ has homogeeous factorsato we have these cocluso as the below: () (): o( ) corme the K ; 3 o( ) q q Cq Cq q a the orers of a q are ot 7 Volume 6 Number 3 July 06
3 3 (): o( ) C C o( ) q q Cq Cq q a are ot corme the K. q a the orers of 3 o( ) o( ) q q Cq Cq q a the orers of a are ot corme the K. q () (): o( ) o( ) q ( q ) the K or q [ ] K q ; [ q] (): o( ) o( ) q ( q) K q [ q ] or K q ]; [ q] (3) (): 3 o( ) o( ) q Cq Cq q a the orers of a are ot corme the K ; (): q [ ] 3 o( ) C C o( ) q a the orers of a are ot corme the K q ; [ q] 4.4. A metho of costructo coectve subgrah: I orer to costruct coectve subgrah we ee M S. We suose { } S m M q a b a b t Ot t m a t j costruct coectve subgrah. a ; ab ; j b j ; ; We have S a a b b M accorg to ths metho we must be I Theorem 4.3 we rove that the comlete grah a the comlete multartte grah of rouct of two fferet rmes ower vertces has abelat homogeeous factorsato. Ths leas to the followg questo. Questo 4.5: Whe ) M q r (v are rmes); r r r 3 3 ) M ( 3 are rmes); Whether the comlete grah a the comlete multartte grah have abelat homogeeous factorzato? Ackowlegmet The authors ackowlege the suort from the Natoal Natural Scece Fouato of Cha uer Grats 630446 the Provcal Scece a Techology Fouato of Guzhou of Cha uer Grat LKA0 a 0500 a the Provcal excellet creatve talets of scece a techology rewar rogram of Guzhou of Cha uer Grats 0455. We woul lke to exress our scere arecato o the aoymous revewers for ther sghtful commets whch have greatly hele us mrovg the 8 Volume 6 Number 3 July 06
qualty of the aer. Refereces [] Guralck R. & L C. H. & Praeger C. E. et al. (004). O orbtal arttos a excetoalty of rmtve ermutato grous Tras. Amer. Math. Soc. 356 4857-487. [] Lm T. K. (003). Ege-trastve homogeeous factorsato of comlete grahs. Ph.D. Thess the Uversty of Wester Australa. [3] Strger L. (004). Homogeeous factorsatos of comlete grahs. M.Sc. Dssertato the Uversty of Wester Australa. [4] Cuaresma M. C. N. (004). Homogeeous factorsatos of Johso grahs. Ph. D. Thess Uversty of the Phles. [5] Guc M. L C. H. Potock P. & Praeger C. E. (007). Homogeeous factorsato of comlete multartte grahs. Dscrete Math 307 45-43. [6] Fag X. G. L C. H. & Wag J. (007). O trastve -factorsatos of arc-trastve grahs. J. Comb. Theory Ser. A. 4 69-703. [7] Guc M. L C. H. Potock P. & Praeger C. E. (006). Homogeeous factorsato of grahs a grahs. Euro. J. Combattcs 7-37. [8] Praeger C. E. L C. H. & Strger L. (009). Commo crculat homogeeous factorsatos of the comlete grah. Dscrete Math 309 3006-30. [9] L C. H. (00). O somorhsms of fte Cayley grahs -a survey. Dscrete Math 56 30-334. Huhog Wu receve the B.S. egree ale mathematcs from Shax Uversty Shax Cha 00 a the M.S. egree ale mathematcs from Yua Uversty Yua Cha 009. She s curretly a assocate rofessor at School of Sceces Ashu Uversty. She has ublshe over 0 joural aers clug comuter alcato comuter egeerg a so o. Her research terests clue grous a grah evolutoary otmzato a ts alcato to Kasack Problems a mathematcal moelg. Shuqu Qa receve the B.S. egree mathematcs a ale mathematcs from FuYa Normal College Ahu Cha 000 a the M.S. egree oeratoal research a cyberetcs from the School of Guzhou Uversty Guzhou Cha 007. He s curretly ursug the Ph.D. egree cotrol theory a cotrol egeerg from the College of Automatc Egeerg Najg Uversty of Aeroautcs a Astroautcs Najg Cha. Shuqu Qa s curretly a assocate rofessor at School of Sceces Ashu Uversty. He has ublshe over 30 joural aers clug IEEE Trasacto o Cyberetcs Cotrol Theory a Alcato a so o. Hs research terest clues mmue otmzato evolutoary otmzato a ts alcato to ower electrocs a mathematcal moelg. 9 Volume 6 Number 3 July 06