Discrete Mathematics

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1 Discrete Mathematics 3 (20) Cotets lists available at ScieceDirect Discrete Mathematics joural homepage: wwwelseviercom/locate/disc The degree sequece of Fiboacci ad Lucas cubes Sadi Klavžar ab Michel Mollard c Marko Petkovšek a a Faculty of Mathematics ad Physics Uiversity of Ljubljaa Sloveia b Faculty of Natural Scieces ad Mathematics Uiversity of Maribor Sloveia c CNRS Uiversité Joseph Fourier Istitut Fourier BP rue des Maths St Marti d Hères Cedex Frace a r t i c l e i f o a b s t r a c t Article history: Received 30 March 200 Received i revised form 3 November 200 Accepted 2 March 20 Available olie 5 April 20 Keywords: Fiboacci cube Lucas cube Degree sequece Geeratig fuctio The Fiboacci cube Γ is the subgraph of the -cube iduced by the biary strigs that cotai o two cosecutive s The Lucas cube Λ is obtaied from Γ by removig vertices that start ad ed with It is proved that the umber of vertices of degree k i ad k i0 Γ ad Λ k 2i i 2i i 2i is i0 i 2i+ i 2i+ i respectively Both results are obtaied i two ways sice each of the approaches yields additioal results o the degree sequeces of these cubes I particular the umber of vertices of high resp low degree i Γ is expressed as a sum of few terms ad the geeratig fuctios are give from which the momets of the degree sequeces of Γ ad Λ are easily computed 20 Elsevier BV All rights reserved 2 + Itroductio A Fiboacci strig is a biary strig that cotais o two cosecutive s The Fiboacci cube Γ 0 is defied as follows Its vertices are all Fiboacci strigs of legth two vertices are adjacet if they differ i precisely oe bit I particular Γ 0 K Γ K 2 ad Γ 2 is the path o three vertices Alteratively Γ ca be defied as the so-called simplex graph of the complemet of the path o vertices cf [] Fiboacci cubes were itroduced as a model for itercoectio etworks [7] ad received a lot of attetio afterwards For differet studies of their structure we refer to [236823] These cubes also foud a applicatio i theoretical chemistry There perfect matchigs i hexagoal graphs reflect the stability of the correspodig bezeoid molecules ad the so-called resoace graphs capture the structure of the perfect matchig It is appealig that Fiboacci cubes are precisely the resoace graphs of a special class of hexagoal graphs called fiboaccees the result proved i [0] We also metio that Fiboacci cubes led to the cocept of the Fiboacci dimesio of a graph [] ad that they ca be recogized i O( E(G) log V(G) ) time [5] Lucas cubes form a class of graphs closely related to Fiboacci cubes The Lucas cube Λ 0 is the subgraph of the -cube iduced by Fiboacci strigs b b such that ot both b ad b are I particular Λ 0 Λ K ad Λ 2 Γ 2 is the path o three vertices For differet aspects of Lucas cubes see [26896] I this paper we are iterested i the degree sequece of Fiboacci ad Lucas cubes Oe of our motivatios is the fact that several partial results were previously obtaied i order to attack differet problems o Fiboacci cubes I the semial paper [7 Lemma 6] it was observed that the degrees of Γ are at least (+2)/3 ad (obviously) ot more tha More tha te years later a recursive formula for computig the degree of ay vertex of Γ is give i [3] It depeds o the recursive structure of Γ ad the value of the iteger that represets the give vertex ( biary umber) This approach was further Correspodig author at: Faculty of Mathematics ad Physics Uiversity of Ljubljaa Sloveia addresses: sadiklavzar@fmfui-ljsi (S Klavžar) michelmollard@ujf-greoblefr (M Mollard) markopetkovsek@fmfui-ljsi (M Petkovšek) X/$ see frot matter 20 Elsevier BV All rights reserved doi:006/jdisc200309

2 S Klavžar et al / Discrete Mathematics 3 (20) developed i [4] where the degrees are used to ivestigate the domiatio umber of the Fiboacci cubes I the mai result o the degrees [4 Theorem 26] vertices of degrees 2 ad 3 are explicitly described However the approach i geeral does ot give the umber of vertices of Γ of a give degree a fudametal property of a give family of graphs For k 0 let f k deote the umber of vertices of Γ havig degree k The our first mai result is: Theorem For all k 0 2i i + f k k i k i + i0 () Note that oly the terms with i betwee ( k)/2 ad mi(k k) are ozero which could be useful whe evaluatig these umbers A aalogous remark holds for the subsequet summatio formulas as well I the ext sectio we prove Theorem by derivig ad solvig a correspodig system of liear recurreces The i Sectio 3 several cosequeces of Theorem are preseted A special emphasis is give o vertices of small ad large degrees For istace Corollary 34 i particular covers the degrees of the above-metioed [4 Theorem 26] I Sectio 4 we give a direct approach to Theorem by cosiderig degrees via the partitio of V(Γ ) ito strigs of a give weight I this way ot oly Theorem is reproved but (i) the vertices of a give degree ad weight are eumerated thus givig additioal iformatio o the Fiboacci semilattice [4] (ad the Lucas semilattice [6]) ad (ii) the way to our secod mai theorem is paved Deotig by l k k 0 the umber of vertices of Λ havig degree k we prove i Sectio 5: Theorem 2 For all k 0 with 2 [ i 2i i l k 2 + 2i + k k i 2i + k i0 2i k i ] (2) Fially i Sectio 6 we reprove Theorem 2 by the method of geeratig fuctios This approach is somewhat more ivolved tha the oe take i Sectio 5 however it ca be further used to obtai several additioal properties of the sequece of degrees of the Fiboacci ad Lucas cubes m Throughout the paper we follow the defiitio of biomial coefficiets give i [5] I particular 0 m ad k 0 for all m k Z with k < 0 We fid this remark importat sice ot all curretly used computer algebra systems follow this covetio 2 Proof of Theorem The vertex set of Γ aturally decomposes ito the sets A ad B cosistig of those strigs that start with a ad those strigs that do ot start with a respectively Hece A 0 B 0 {λ} (where λ is the empty strig) ad for A {α α B } ad B {0α α A B } Sice every vertex i A 2 ecessarily starts with 0 A iduces Γ 2 i Γ O the other had B iduces Γ i Γ Moreover each vertex α of A has exactly oe eighbor i B amely 0α We ow give the key defiitio that will eable us to compute the degree sequece of Γ For ay ad ay 0 k let a k respectively b k be the umber of vertices of A respectively B of degree k Cosider a vertex x A of degree k The it is of degree k i the subgraph Γ 2 of Γ iduced by A Sice x lies i exactly oe of the correspodig sets A 2 ad B 2 we get a k a 2 + b 2 Similarly a vertex y B either has a eighbor i A (if it starts with 00) or has o eighbor i A I the first case it is a vertex of the correspodig set B i the secod case a vertex of A Therefore b k b + a k Hece the degree sequeces i the subgraphs iduced by A ad B satisfy the system of liear recurreces ad iitial coditios a k a 2 + b 2 ( 2 k ) (3) b k b + a k ( k ) a 0k a 0 0 ( 0 k 0) a a k 0 (k 2) b 00 b 0k b 0 0 ( k ) (4)

3 32 S Klavžar et al / Discrete Mathematics 3 (20) Their geeratig fuctios a(x y) k 0 a kx y k ad b(x y) k 0 b kx y k therefore satisfy the system of liear algebraic equatios a(x y) xy x 2 ya(x y) + x 2 yb(x y) b(x y) xyb(x y) + xa(x y) whose solutio is xy( + x xy) a(x y) ( xy)( x 2 y) x 3 y b(x y) ( xy)( x 2 y) x 3 y Write u xy v x 2 y The b(x y) uv x 3 y (uv) x 3 y(uv) h 0 (xy) h (x 2 y) h ( xy) h+ ( x 2 y) h 0 i j x i+2j y i+j h h h h 0 ij h h+ y h I the last step we used the well-kow expasio x k ( x) i x i k+ k i k x 3h y h (uv) h Now replace summatio variables h ad i by i + 2j ad k i + j h The i 2j ad h k j so b(x y) 2j j x y k k j k j kj 0 (5) hece b k 2j j0 k j j k j (6) From (3) ad (4) (or alteratively from (5)) we obtai a k b b 22 + b 2 ( 2 k 2) (7) Deote 2j j t kj k j k j The [ ] 2j 2j 2 j t j t 22j + t 2j + t 2j k j k j 2 k j 2j 2 j 2j 2 j + k j k j k j k j 2j 2 j + k j k j by usig Pascal s idetity twice hece it follows from (6) ad (7) that 2j 2 j + a k k j k j j 0 2j j k j k j + j0 ( 2 k 2) (8)

4 S Klavžar et al / Discrete Mathematics 3 (20) Here we replaced j by j 0 ad oted that 0 for It is easy to check that (8) holds whe k {0 } k k+ or {0 } as well Fially from (6) ad (8) we obtai 2j j + f k a k + b k k j k j + j0 by usig Pascal s idetity oce more 3 Cosequeces of Theorem Let F be the th Fiboacci umber: F 0 0 F F F + F 2 for 2 Sice V(Γ ) F +2 Theorem immediately implies: Corollary 3 For ay 0 F +2 k0 i0 i + k i + 2i k i (9) We ext give a alterative proof (avoidig Fiboacci cubes) of Corollary 3 Set F() If k > ad i the k i + < 0 thus 0 If k > ad i 0 or if i > k the k0 k i0 2i i 2i i 0 Thus after iterchagig the order of summatio ad usig Vadermode s covolutio i + 2i i + F() k i + k i 2i + i0 k0 i0 Sice for i > ( + )/2 we have 0 we ca restrict our summatio rage to say 0 i + ad obtai i0 2 + i + + i + F() 2i + i i0 where the last equality holds because i + 0 Usig the well-kow idetity m i0 m i F i m+ see [5 p 289 Eq (630)] we coclude that F() F +2 A iterestig problem useful for applicatios such as domiatio or colorig is to determie for some fixed iteger m the umber of vertices of degree (Γ ) m ad δ(γ ) + m (As usually ad δ deote the maximum ad the miimum degree) Whe m is small the followig two corollaries show that i both cases almost all the terms i the sum of Theorem vaish Corollary 32 For 0 m f m m+ i m/2 2i m i i + m i + (0) Proof By Theorem m 2i i + f m m i m i + i0 If m i + < 0 we have 0 thus we ca assume that i m + If 2i < m we have m i + > i + ad agai m 0 m Corollary 33 Let δ δ(γ ) +2 3 For > 0 ad m δ δ+m f δ+m iδ m 2 2i δ + m i i + δ + m + 2i

5 34 S Klavžar et al / Discrete Mathematics 3 (20) Proof We have i + > 0 thus k+2i i δ m 2 2 the δ + m + 2i 3δ + m 2 m 2 Rewrite the sum i Theorem for k δ + m ad observe that if 4 3δ 3 Hece i this case 0 δ+m+2i Our last two results i particular give the asymptotic behavior of the umber of vertices of degrees (Γ ) m ad δ(γ ) + m whe Corollary 34 Let m 0 ad let 2m + 2 The ; m 0 2; m + ; m 2 f m 3 8; m 3 2 /2 + 3/2 2; m ; m 5 More geerally f m is a polyomial i of degree m/2 Its leadig coefficiet is is odd Proof Whe i m + ad 2m + 2 we have 2i 0 thus 2i m i m/2 + whe m is eve ad whe m (m/2)! m/2! 2i m i Hece havig i mid Corollary 32 the first values are thus obtaied directly from m 2i i + m i m i + i m/2 2i Cosider ow this sum for some fixed m For all i is a polyomial i with leadig term m i m i (m i)! ad is m idepedet of Thus f m is a polyomial i Its leadig moomial is obtaied from the term correspodig to the miimal i such that 0 which is equivalet to 2i m ad further to i m/2 Hece the miimal such that i is m m/2 ad deg f m m m/2 m/2 If m is eve the whe i m/2 m/2 thus the leadig term is m (m/2)! m/2 If m is odd the m/2 + whe i m/2 thus i this case the leadig term is m/2 + m/2 m m/2! Corollary 35 Let δ δ(γ ) +2 3 The ; m 0 3p (p + )(p + 4); m 0 3p + 2 p + 2; m 0 3p + 2 f δ+m p(p + )(p + 8); m 3p 6 20 p(p + )(p3 + 24p 2 + 8p + 4); m 3p + 24 (p + )(p + 2)(p2 + 5p + 2); m 3p + 2 More geerally for all m 0 f δ+m is: a polyomial i p of degree 3m ad leadig coefficiet a polyomial i p of degree 3m + 2 ad leadig coefficiet a polyomial i p of degree 3m + ad leadig coefficiet (3m)! (3m+2)! for 3p; for 3p + ; for 3p + 2 (3m+)! Proof The first values are obtaied by direct use of Corollary 33 Let m be fixed ad cosider the geeral case whe 3p for some fixed m The δ p ad by itroducig a ew summatio variable j i p we ca rewrite the sum of Corollary 33 as m p 2j p + j + f 3pp+m m j 2j + m j m 2

6 S Klavžar et al / Discrete Mathematics 3 (20) p 2j p+j+ Notice that is a polyomial i p of degree m j ad is a polyomial i p of degree 2j + m therefore their m j 2j+m p 2j product is of degree 2m + j The maximum degree will be obtaied whe j is maximum ie j m The ad m j p+j+ p+m+ thus the leadig term is p3m 2j+m 3m (3m)! The cases 3p + ad 3p + 2 are treated similarly The maximum degree isobtaied whe j is maximum which p 2j+ p+j+ p+m+2 i these two cases is j m + Whe 3p + we have ad thus the leadig term m j+ 2j+m 3m+2 is p3m+2 (3m+2)! Whe 3p + 2 the p 2j+2 p+j+ p+m+2 ad thus the leadig term is p3m+ m j+ 2j+m 3m+ (3m+)! 4 Eumeratio of vertices i Γ by weight The purpose of this sectio is to determie the umber of vertices i Γ with a give weight ad degree where the weight of a biary strig is the umber of s i it This could be doe by meas of geeratig fuctios as i Sectio 2 evertheless we use a direct approach which alog the way gives some additioal iformatio about Fiboacci strigs As a cosequece we are able to give a alterative proof of Theorem as well as a combiatorial iterpretatio of the summatio expressio From this approach we ca also describe easily the set of vertices of a give weight ad degree ad deduce quickly the degree sequece of Lucas cubes We leave the latter task for the ext sectio ad cotiue here with the study of the structure of Fiboacci strigs For 0 deote F the set of all Fiboacci strigs of legth L the set of all Lucas strigs of legth S ij {α F ; α starts with i ad eds with j} i j {0 } where S 00 also icludes the empty strig λ Note that i the otatio of Sectio 2 S S 0 A ad S 0 S 00 B I additio for ay iteger m 0 we itroduce the followig Fiboacci strigs: α m (0) m 0 β m (0) m γ m (0) m δ m (0) m We call the strigs δ m degeerate Fiboacci strigs Lemma 4 Every odegeerate Fiboacci strig ca be uiquely decomposed as β m0 0 l 0 α m 0 l α m2 0 l 2 α mp 0 l p γ mp+ where p 0 l 0 l p 0 m m p ad m 0 m p+ 0 Moreover m 0 ad m p+ determie to which of the sets S S0 or S00 the strig belogs S0 Proof The proof of the existece of such a decompositio is by iductio o the legth of the strig This is clearly true for strigs of legth 2 Cosider ow a strig s of legth > 2 Suppose first that s 0s where s F By iductio we have the followig possibilities for s : s β m0 0 l 0 αm 0 l αmp 0 l p γ mp+ hece s α m0 0 l 0 αm 0 l αmp 0 l p γ mp+ ; s 0 l 0 αm 0 l αmp 0 l p γ mp+ hece s 0 l 0+ α m 0 l αmp 0 l p γ mp+ ; s α m 0 l αmp 0 l p γ mp+ hece s 0α m 0 l αmp 0 l p γ mp+ ; s γ m thus ow s 0γ m ; s δ m ad hece s γ m + Similarly if s s s F we have the followig cases: s 0 l 0 αm 0 l αmp 0 l p γ mp+ hece s β 0 l 0 α m 0 l αmp 0 l p γ mp+ ; s α m 0 l αm2 0 l 2 αmp 0 l p γ mp+ hece s β m +0 l αm2 0 l 2 αmp 0 l p γ mp+ ; s γ m but the s δ m is degeerate Hece i each of the cases we have obtaied a decompositio of s i the expected form It is immediate to verify that strigs from S satisfy m 0 > 0 ad m p+ > 0; strigs from S 0 satisfy m 0 > 0 ad m p+ 0; strigs from S 0 satisfy m 0 0 ad m p+ > 0; ad strigs from S 00 satisfy m 0 0 ad m p+ 0 To prove uiqueess cosider first a strig β m0 0 l 0 αm 0 l αmp 0 l p γ mp+ from S thus with m 0 > 0 ad m p+ > 0 I the three possible cases (l 0 > 0 l 0 0 ad m > 0 p 0) such a strig cotais at least two cosecutive 0 s so the strig is ot degeerate O the other had it is clear that a odegeerate strig caot be decomposed i two ways as β m0 0 l 0 αm 0 l αmp 0 l p γ mp+

7 36 S Klavžar et al / Discrete Mathematics 3 (20) Note also that the degeerate Fiboacci strig δ m is of legth 2m + weight w m + ad the correspodig vertex of Γ is of degree k m + For all the other strigs we have: Propositio 42 A Fiboacci strig β m0 0 l 0 αm 0 l αm2 0 l 2 αmp 0 l p γ mp+ is of legth p i0 l i + p+ 2 i0 m i + p ad weight w p+ i0 m i ad the correspodig vertex of Γ is of degree k p i0 l i + p+ i0 m i Proof The assertio for the legth ad the weight follows immediately from defiitios As for the degree use the fact that chagig a to 0 i a vertex from F gives a vertex i F while a 0 ca be chaged to oly if it is ot adjacet to ad thus ot iside a block of the form α m β m ad γ m We will use the followig classical results about the compositio of itegers Lemma 43 Let a b 0 The the umber of solutios of x + x x a b with x x 2 x a oegative itegers is b+a b Lemma 44 Let a b 0 The the umber of solutios of x + x 2 + +x a b with x x 2 x a positive itegers is I the rest we will use some more otatio Let s k s0 k s0 k ad s00 k be the umber of vertices of degree k i S S0 S0 ad S00 respectively Let i additio S w S0 w S0 w ad S00 w be the correspodig sets where each vertex is of weight w ad let s kw s0 kw s0 kw ad s00 kw be the umber of vertices of degree k i these sets respectively Lemma 45 For all itegers k w w s 00 kw 2w + k kw s 0 kw s0 s kw w 2w + k 2 2w k w w 2w 2w + k k w 2w k w Proof Assume first that w k A strig from Sw 00 0 αm 0 l αmp 0 l p where p 0 l 0 l l p 0 ad m m p > 0 By Propositio 42 there is a mappig betwee Sw 00 ad the solutios of p k w 0 l l p k w with l 0 l p 0 () m + + m p w with m m p A strig from Sw 0 m 0 0 l 0 αm 0 l αm2 0 l 2 αmp 0 l p with p 0 l 0 l l p 0 m 0 m m p > 0 Thus there is a - mappig betwee Sw 0 ad the solutios of p k w 0 l l p k w with l 0 l p 0 (2) m m p w with m 0 m p A odegeerate strig from Sw m 0 0 l 0 αm 0 l αm2 0 l 2 αmp 0 l p γ mp+ where p 0 l 0 l p 0 m 0 m p > 0 Thus there is a - mappig betwee these strigs ad the solutios of p k w 0 l l p k w with l 0 l p 0 (3) m m p+ w with m 0 m p+ b b a

8 S Klavžar et al / Discrete Mathematics 3 (20) w 2w+ w Assume that p w w 0 the by Lemmas 43 ad 44 the umber of solutios of () (3) are w 2w+ 2w w ad w 2w+ 2 2w w respectively Assume ow that k w < 0 The there are o solutios of () (3) thus there are o odegeerate strigs of degree k i Sw 00 S0 w ad S w Notice that we have w because w 0 implies k < 0 a cotradictio Suppose k w 2 The we ca write 2w + k > 2w + k > 2w + k 2 w + (w + k 2) w > w 0 0 w thus 2w+ w 2w+ Assume that k w The w 2w+ 2 2w + k > 2w + k w > w 0 w w w 2w therefore 0 Cosider ow w 2w 2w w 2w+ 2w+ 2w+ 2 w w w w w This umber is zero if k > w Otherwise (if k w ad 2k ) it is which correspods to the degeerate strig δ By symmetry we have s 0 kw s0 kw A vertex of weight w has degree at least k thus there are o vertices of degree k i the sets Sw S0 w S0 w S00 w if w k is ot satisfied It is immediately verified that the four formulas also hold Let f kw be the umber of vertices of Γ havig degree k ad weight w The we have: Theorem 46 For all itegers k w with k w w + 2w f kw w k + k w Proof Clearly f kw s kw +s0 kw +s0 kw +s00 kw Applyig Lemma 45 ad (three times) the idetity a b + a b a+ b we arrive at w + 2w f kw 2w + k k w Because w + > w+ 0 we have 2w+ w+ w k+ Note that by the covetio we are usig for the biomial coefficiets f kw 0 whe w > ( + )/2 Theorem immediately follows from Theorem 46 5 Proof of Theorem 2 Let l kw be the umber of vertices of Λ of degree k ad weight w ad let l pq kw for p q {0 } be the umber of such strigs i the set S pq Lemma 5 For all k w such that 2 k ad 0 w l 00 kw s00 w + s0 kw l 0 kw l0 kw s0 kw + s kw l kw 0 Proof A Lucas strig that starts ad eds with 0 ca be writte as 0s where either s S 00 w is of degree or s S0 w is of degree k This gives the first equality Similarly we obtai the secod equality while the last oe is obvious Theorem 52 For all k w such that k w 0 ad 2 w 2w w 2w l kw + 2 (4) 2w + k k w 2w + k k w

9 38 S Klavžar et al / Discrete Mathematics 3 (20) Proof Assume first that k Sice l kw l 00 kw + 2l0 kw Lemmas 45 ad 5 imply that w 2w w 2w l kw + 2w + k k w 2w + k k w w 2w w 2w w + k k w 2w + k k w Usig Pascal s idetity we ca group the first term with oe half of the third term the secod term with oe half of the fourth term ad the remaiig half of the third term with the remaiig half of the fourth term to obtai (4) The oly Lucas strigs of degree k 0 are λ ad 0 hece l 0w 0 whe 2 But i this case the right-had side of (4) evaluates to 0 as well Note agai that by the covetio we are usig for the biomial coefficiets l kw 0 whe w > /2 Theorem 2 ow follows immediately from Theorem 52 Corollary 53 Let The umber of vertices of weight w i L is w w l kw + w 2w k0 Proof Note first that the result is true whe w 0 or Assume ow that w ad 2 The w w ad w 2w+ w w Hece we obtai from Theorem 52 by Vadermode s covolutio [ ] w 2w w 2w l kw + 2 k w k w k w k w k0 k0 w w + 2 2w 2w w Usig Pascal s idetity ad 2w w w we have the fial expressio w 2w+ Similarly as Theorem yields special cases for specific degrees i Fiboacci cubes oe ca apply Theorem 2 to obtai the umber of vertices of certai degrees i Lucas cubes For istace l ( 2) l 0( 3) ad l 2 ( 5) For the miimal degree if 2 the 3; 0(mod 3) l (+2)/3 ( + 5)/6; (mod 3) ; 2(mod 3) 6 The method of geeratig fuctios I this sectio we approach Theorem 2 usig geeratig fuctios It is relatively more complicated tha the approach from the previous two sectios O the other had it eables us to obtai may additioal results as demostrated at the ed of the sectio by several examples Clearly F S S 0 S 0 S 00 (for 0) (5) L S 0 S 0 S 00 (for 0) S 0F 4 0 (for 4) S 0 0F 3 0 (for 3) S 0 0F 3 0 (for 3) S 00 0F 2 0 (for 2) Eq (5) shows that V(Γ ) F ca be partitioed ito four blocks which by (7) (20) iduce i Γ with 4 a Γ 4 a Γ 3 a Γ 3 ad a Γ 2 respectively By (5) agai each of these blocks ca be further partitioed ito four subblocks S 0S 40 0S0 40 0S0 40 0S00 40 (2) (6) (7) (8) (9) (20)

10 S Klavžar et al / Discrete Mathematics 3 (20) Fig Perfect matchigs betwee subblocks ad uios of subblocks of Γ S 0 0S 3 0 0S S S00 30 (22) S 0 0S 30 0S0 30 0S0 30 0S00 30 (23) S 00 0S 2 0 0S S S00 20 (24) Propositio 6 The set of those edges of Γ ot cotaied withi oe of the four blocks i (5) equals 8 i M i where each M i is a perfect matchig betwee a subblock ad the uio of a pair of subblocks of differet blocks as follows (see Fig ): M is a perfect matchig betwee 0S ad 0S S0 2 M 2 is a perfect matchig betwee 0S 0 20 ad 0S0 30 0S 3 M 3 is a perfect matchig betwee 0S M 4 is a perfect matchig betwee 0S M 5 is a perfect matchig betwee 0S M 6 is a perfect matchig betwee 0S M 7 is a perfect matchig betwee 0S M 8 is a perfect matchig betwee 0S ad 0S00 0 0S0 4 0 ad 0S0 0 0S ad 0S S0 0 ad 0S S ad 0S00 0 0S0 4 0 ad 0S0 0 0S Proof We eed to aalyze the exteral coectios of each of the 6 subblocks of Γ By way of example we do this for the subblock 0S 0 30 i all the other cases the aalysis is similar Each strig σ 0S0 30 is of the form σ 0τ 00 where τ F 5 So σ is adjacet to precisely oe vertex 0τ 0 S (if τ eds with the 0τ 0 0S 40 otherwise 0τ 0 0S0 4 0); o vertices i S 0 sice each vertex of S0 is at a distace 2 or more from each vertex of S 0 ; precisely oe vertex i S 00 amely 00τ 00 0S Whe aalyzig other subblocks we fid out i a similar way that each vertex i 0S 40 0S0 40 is adjacet to precisely oe vertex i 0S0 each vertex i 0S is adjacet to precisely oe vertex i 0S00 2 0; each vertex i 0S is adjacet to precisely oe vertex i 0S S Take together these facts imply that the exteral coectios of the subblock 0S are precisely the edges of M 5 M 8 with oe edpoit i 0S ;

11 320 S Klavžar et al / Discrete Mathematics 3 (20) It follows from (2) (24) ad from Propositio 6 that s k s 42 + s s s00 42 ( 4 k 2) s 0 k s 3 + s s0 3 + s00 32 ( 3 k 2) s 0 k s 3 + s0 3 + s s00 32 ( 3 k 2) s 00 k s 2k + s0 2 + s0 2 + s ( 2 k 2) Together with the correspodig iitial coditios this system of recurreces implies that the geeratig fuctios s (x y) s k x y k k 0 s 0 (x y) k 0 s 0 (x y) k 0 s 00 (x y) k 0 s 0 k x y k s 0 k x y k s 00 k x y k satisfy the system of liear algebraic equatios s (x y) xy + x 3 y 2 + x 4 y 2 (s (x y) + s 0 (x y) + s 0 (x y) + s 00 (x y)) s 0 (x y) x 2 y + x 3 y(s (x y) + s 0 (x y)) + x 3 y 2 (s 0 (x y) + s 00 (x y)) s 0 (x y) x 2 y + x 3 y(s (x y) + s 0 (x y)) + x 3 y 2 (s 0 (x y) + s 00 (x y)) s 00 (x y) + xy + x 2 s (x y) + x 2 y(s 0 (x y) + s 0 (x y)) + x 2 y 2 s 00 (x y) whose solutio is s (x y) xy( xy) ( xy)( x 2 y) x 3 y s 0 (x y) s 0 x 2 y (x y) ( xy)( x 2 y) x 3 y (26) s 00 (x y) x 2 y ( xy)( x 2 y) x 3 y Expadig these ratioal fuctios ito power series we obtai s w 2w k 2w + k 2 k w w0 s 0 k s0 w 2w k 2w + k k w w0 w 2w s 00 k 2w + k k w w0 By otig that f k s + k s0 + k s0 + k s00 k ad by usig Pascal s idetity repeatedly we obtai () agai To recompute l k ote that for 3 L 0F 3 0 0F 0F 3 0 (0S 0S0 0S0 0S00 ) Each σ 0F 3 0 is of the form σ 0τ 0 with τ F 3 Hece σ is adjacet to precisely oe vertex i 0F amely 00τ 0 0S 00 Coversely each vertex 00τ 0 0S00 is adjacet to 0τ 0 0F 30 So for 3 k l k f 3 + s k + s0 k + s0 k + s00 f 3 + f k + s 00 s00 k (29) Usig () ad (28) this formula ca be show equivalet to (2) (25) (27) (28)

12 S Klavžar et al / Discrete Mathematics 3 (20) From (25) (29) ad the values l 00 l 0 l 0 it is straightforward to compute the geeratig fuctios f (x y) f k x y k + xy + ( y)x2 y ( xy)( x 2 y) x 3 y k 0 l(x y) k 0 l k x y k + ( y)x + x2 y 2 + ( y)x 3 y ( y) 2 x 4 y ( xy)( x 2 y) x 3 y from which additioal iterestig iformatio cocerig the degree sequeces f k k0 ad l k k0 ca be obtaied easily For istace: Sice the geeratig fuctios f (x y) l(x y) s (x y) s 0 (x y) s 0 (x y) s 00 (x y) all have ( xy)( x 2 y) x 3 y xy x 2 y 2 x 3 y + x 3 y 2 as their deomiator each of the sequeces s k {f k l k s k s0 k s0 k s00 k } satisfies the same recurrece s k s + s 2 + s 3 s 32 for all large eough ad k 2 From 0 x f k f (x y) y + x x x F 2 +2 x 0 k0 it follows that V(Γ ) F +2 ad from x l k l(x y) y + x2 x x L 2 x 0 0 k0 it follows that V(Λ 0 ) L 0 V(Λ ) L for 3 From 0 x kf k y f (x y) y k0 2x ( x x 2 ) F + + 2( + )F it follows that E(Γ ) (F + + 2( + )F )/5 ad from x kl k y l(x y) 2(2 x)x2 y ( x x 2 ) 2F 2 x 0 k0 0 it follows that E(Λ ) F 5 4 More geerally for each p 0 oe ca easily compute the geeratig fuctios of the sequeces of the p-th momets k0 kp f k resp 0 k0 kp l k of the degree sequeces f 0 k k0 resp l k k0 from the higher derivatives of f (x y) resp l(x y) Sice p y f (x y) p y x k p f k 0 k0 where k p p j0 (k j) is the p-th fallig power of k we have x k p f k p x S pj k j f k 0 k0 0 k0 j0 p S pj j0 0 x k j f k k0 p j0 x S pj j y j f (x y) y where S pj deotes Stirlig umbers of the secod kid Similarly 0 x k p l k k0 p j0 S pj j y j l(x y) y

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