A Simple Research of Divisor Graphs

Transcription

1 The 29th Workshop on Combnatoral Mathematcs and Computaton Theory A Smple Research o Dvsor Graphs Yu-png Tsao General Educaton Center Chna Unversty o Technology Tape Tawan yp-tsao@cuteedutw Tape Tawan 116 Republc o Chna Abstract Let S be a nte nonempty set o postve ntegers Then the dvsor graph D (S) o S has S as ts verte set and vertces and j are adjacent and only ether j or j Ths paper nvestgates the verte-chromatc number the clque number the clque cover number and the ndependence number o D ([n]) and ts complement where [ n ] { :1 n n Ν} Besdes we dscuss the perect property on ths knd o graphs In the last secton we also gve some notons about the bandwdth o the dvsor graphs 1 Introducton In 2000 Sngh and Santhosh [4] dened the concept o a dvsor graph They dened a dvsor graph G as an ordered par ( V E) where V Ζ and or all u v V u v uv E and only u v or v u Sngh and Santhosh [4] showed that every odd cycle o length ve or more s not a dvsor graph whle all even cycles complete graphs and caterpllars are dvsor graphs In 2001 Chartrand Muntean Saenpholphant and Zhang [1] also studed dvsor graphs They let S be a nte nonempty set o postve ntegers Then the dvsor graph D (S) o S has S as ts verte set and vertces and j are adjacent and only ether j or j A graph G s a dvsor graph G S) or some nonempty nte set S o postve ntegers Hence G s a dvsor graph then there ests a uncton : V ( G) Ν called a dvsor labelng o G such that G ( V ( G))) The results o [4] are conrmed n [1] where t was shown that trees and bpartte graphs are dvsor graphs and a characterzaton o all dvsor graphs was gven Moreover Le Anh Vnh [3] showed that or any postve nteger n m s an nteger between 0 and n then there ests a dvsor graph * Ths work was supported n part by the Natonal Scence Councl o the Republc o Chna under Contract NSC M o order n and sze m Chrstopher Frayer [2] gave a condton or Cartesan product o graphs such that t could be a dvsor graph We devote the man o ths artcle to the verte -chromatc number the clque number the clque cover number the ndependence number and the bandwdth o D ([n]) where [n] means the set { n 1 n} For convenence D ([n]) and ts complement are wrtten smply as D ( and D ( later respectvely 2 Chromatc number and clque number The chromatc number o a graph G wrtten χ (G) s the mnmum number o colors need to label the vertces so that the adjacent vertces receve derent colors The clque number o a graph G wrtten ω (G) s the mamum sze o a set o parwse adjacent vertces (clque) n G The ollowng lemma s trval Lemma 1 For any graph G ω( G) χ( G) Theorem 2 χ ( log2 n + 1 ω( Proo At rst we gve an upper bound o χ ( wth the mappng c : V ( { :1 log2 n + 1} dened by c ( ) log2 + 1 Let c ( ) c( y) then log2 + 1 log2 y + 1 mples log2 log2 y < 1 e 1 2 < y < 2 and so y otherwse s not adjacent to y because s not a dvsor and a multple o y Snce c s a proper log 2 n + 1 colorng χ ( log2 n + 1 Net we show that log 2 n + 1 s a lower bound o ω ( Clearly { 2 : 0 log 2 n} s a clque It s easy to get ω ( log2 n + 1 From Lemma 1 we have log2 n + 1 ω( χ( log2 n

2 The 29th Workshop on Combnatoral Mathematcs and Computaton Theory and the equalty s obtaned Theorem 3 χ ( ) n ω( ) Proo At rst we gve an upper bound o χ ( ) wth the mappng c : V ( ) { :1 n } j dened by c ( ) where ( 2 1) 2 Let c ( ) c( y) then y and so y otherwse s not adjacent to y because s a proper dvsor or a proper multple o y Snce c s a proper n colorng χ ( ) n Net we show that n s a lower bound o ω ( ) For 6 k 3 n 6k + 2 k Ν let C ( be { : 2k 4k 1} { 2 1: 2k + 1 n } then t s easy to check that C ( s a clque o sze n Thus ω ( ) n By Lemma 1 we have n ω ( ) χ( ) n and the equalty s obtaned 3 Clque cover number and ndependence number A clque cover o G s a partton o V (G) nto clques The mnmum number o clques n a clque cover o G s called the clque cover number o G and s denoted by θ (G) An ndependent set (or stable set) n a graph s a set o parwse nonadjacent vertces The ndependence number o a graph G denoted by α (G) s the mamum sze o an ndependent set o vertces Note that the ollowng lemmas are well-known Lemma 4 For any graph G θ ( G) χ( G) Lemma 5 For any graph G α ( G) ω( G) Theorem 6 (1) θ ( n α( θ log2 n + 1 α (2) ( ) ( ) Proo (1) A straghtorward applcaton o Lemma 4 Lemma 5 and Theorem 3 (2) A straghtorward applcaton o Lemma 4 Lemma 5 and Theorem 2 In vrtue o the corroboraton o Theorem 2 t s natural that we wll ask a queston: Is D ( a perect graph? We say that graph G s a perect ω ( H ) χ( H ) or each ts nduced subgraph H Theorem 7 D ( s a perect graph Proo For Ν postve actors o such that j F () F' s and let F be a subset o some j or j or Γ denotes the set o all these F denotes a mamum F n Γ () Let H be an nduced subgraph o D ( We dene a mappng c rom V (H ) to Ν by c ( ) F or V (H ) Let be adjacent to j WLOG assume j F then { j} Γ( j) and Snce F { j} Fj F j c( ) F < F { j} c( j) and hence c s a proper colorng Let V ( H ) ρ ma F then χ(h ) ρ Let F be some set havng ρ elements Because F y s a clque n H χ( H ) ρ ω( H ) By Lemma 1 we have ω ( H ) χ( H ) Ths completes the proo Corollary 8 A dvsor graph s a perect graph Proo In act a dvsor graph s an nduced subgraph o some D ( Snce each nduced subgraph o ths dvsor graph also s an nduced subgraph o D ( wth the same argument n Theorem 7 we derve the result 4 Bandwdth We call a numberng o a graph G s a bjecton rom V (G) to [ V (G) ] When the vertces o a graph G are numbered wth dstnct ntegers the dlaton s the mamum derence between ntegers assgned to adjacent vertces The bandwdth B (G) o a graph G s the mnmum dlaton o a numberng o G That s to say the dlaton o on G wrtten y 187

3 The 29th Workshop on Combnatoral Mathematcs and Computaton Theory Dl (G) (or (G) ) s dened as B ma{ ( u) ( v) : uv E( G)} and the bandwdth o G s dened as B( G) mn{ B ( G) : s a numberng o G } In ths secton we ocus the study on the bandwdth o D (S) or S [n] For a start we ntroduce two trval lemmas Lemma 9 I H s a subgraph o G then B( H ) ( G) Lemma 10 I G s a complete graph o order n then B ( G) n 1 Lemma 11 I S 2 gcd S mn S then Proo Let be a numberng o D (S) and mn S k Snce gcd S k k s adjacent to each verte n S {k} Thereore we have ( ma{ : j E( } B ma{ ( k) ma{ ( k) 1 S S Because ( S ( j) : j S { k}} ( k)} B s always true or each numberng o D (S) we know that S s a lower bound o Theorem 12 n Proo Snce gcd[ n ] 1 mn[ n] by Lemma 9 t s easy to see B( n Therenater we show that n s also an upper bound Let be a uncton dened on [n] by 1 ( 1) Ater checkng careully we can make sure that s a numberng on [n] Let Ε and Ο be the set o postve even numbers and odd numbers respectvely And let be adjacent to j (WLO G assume j k k Ν ) I k Ο then j 2 2 n I k Ε and Ο then + 1 k I k Ε and Ε then k n n n 2 when s In a word adjacent to j Thus we have ( n and hence B( n B Then we study the condtons leadng to S 2 We call Ω a sel-contaned subset o S Ω s a set contaned n S {1} such that we have N () {1 } Ω or each Ω Let X S we use smply N (X ) to denote the set o vertces outsde n S X havng a neghbor n X Theorem 13 Let S Ν be a nte set contanng 1 Then each o the ollowng states S 2 wll result n (1) There s a sel-contaned subset Ω o S Ω S 2 such that (2) There s a mamum sel-contaned subset Ω Ω < S 2 o S such that (3) There s a sel-contaned subset Ω o S Ω > S 2 and there s a subset such that X o Ω wth X Ω S N ( X ) 2S Ω such that Proo Snce gcd S 1 mn S by Lemma 7 t s clear to see B( S Net we show that S s also an upper bound (1) Let be a one-to-one uncton rom S to [ S ] such that ( Ω) { :1 S Ν} and ( 1) S + 1 Let s adjacent to j or j 1 then j Ω or j Ω by the assumpton Regardless o any case t s easy to check that S B( ( S 2 Thus we obtan 188

4 The 29th Workshop on Combnatoral Mathematcs and Computaton Theory (2) Let be a one-to-one uncton rom S to [ S ] such that ( Ω) { :1 Ω Ν} and ( 1) S + 1 Let s adjacent to j or j 1 then j Ω by the assumpton It s obvous to carry o S 2 Thus we obtan B( ( S 2 (3) Let be a one-to-one uncton rom S to [ S ] such that [ Ω X N ( X ) ] ( Ω X N ( X )) (1) S + 1 ( X N ( X )) Let s adjacent to j or j 1 then j Ω X j X N ( X ) or j Ω by the assumpton It s not dcult to check that S 2 Thereore we derve B( ( S [ Ω + 1] [ Ω X N ( X ) ] In the ollowng we gve three eamples to correspond wth three sucent condtons n the above theorem respectvely (1) Let S { } Then we can nd Ω { } o order 9 19 And hence the numberng o D (S) dened n the table below may prove () () (2) Let S { } Then we can nd a mamum Ω { } o order 8 < S 19 9 And hence the numberng o D (S) dened n the table below may prove () () (3) Let S { } Then we can nd a Ω { } o order 11 > S 19 9 and a X { 4244} wth N (X ) φ where X 2 Ω S N ( X ) 0 7 2S Ω And hence the numberng o D (S) dened n the table below may prove () () For n d Ν let A( n d ) { a : a 1 + ( 1) d [ n]} We attempt to testy A( n d ))) n by applyng Theorem 13 B or d Ν Corollary 14 ( A( n d ))) n From the results o Theorem 10 and Theorem 11 o course we wll surmse the ollowng conjecture s rght: Let S Ν be a nte set contanng 1 then S We ound a countereample easly as ollows: Let S { } Snce { } S s a clque o order 7 n D (S) by Lemma 9 and Lemma 10 we get > S 2 In the nalty or bandwdth o dvsor graphs we gve an easy outcome to clary the wrong guess Proposton 15 Gven any m Ν {1 } or m k m 1 there s a nte subset S o Ν contanng 1 such that S m and k 189

5 The 29th Workshop on Combnatoral Mathematcs and Computaton Theory Proo Let A { 2 : k Ν {0} } and B be a set o m k 1 odd numbers Then t could be vered that A B s the set S what we need Frst at all snce A s a clque o order k + 1 we get A 1 k rom Lemma 9 and Lemma 10 Then we look or an upper bound Let be a one-to-one S such that ( A {1}) [ k 1] (1) k + 1 Let s adjacent to j or j 1 then j A or j B by the assumpton Regardless o any case t s easy to check that k Consequently we obtan B( k And the equalty s acqured uncton rom S to [ ] 5 Concluson and uture works Dvsor graphs are specal and look lke easly to begn Ths work has provded some results on dvsor graphs We thnk that t wll be nterestng or eplorng the other parameters o D ( such as the mamum sze o matchng arborcty prole crcumerence edge-chromatc number acyclc chromatc number acyclc chromatc nde etc Reerences [1] Western G Chartrand R Muntean V Saenpholphat and P Zhang Whch Graphs Are Dvsor Graphs? Congr Numer 151(2001) [2] Chrstopher Frayer Propertes o Dvsor Graphs [3] Le Anh Vnh Dvsor graphs have arbtrary order and sze AWOCA 2006 [4] G S Sngh and G Santhosh Dvsor Graphs-I Preprnt (2000) 190