Discrete Mathematics. Independence polynomials of circulants with an application to music

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1 Discree Mahemaics 309 ( Coes liss available a ScieceDirec Discree Mahemaics joural homepage: wwwelseviercom/locae/disc Idepedece polyomials of circulas wih a applicaio o music Jaso Brow, Richard Hoshio Deparme of Mahemaics ad Saisics, Dalhousie Uiversiy, Halifax, Nova Scoia, Caada B3H 3J5 a r i c l e i f o a b s r a c Aricle hisory: Received 20 Jue 2007 Acceped 13 May 2008 Available olie 11 Sepember 2008 Keywords: Idepedece polyomial Circula Powers of cycles Music The idepedece polyomial of a graph G is he geeraig fucio I(G, x = 0 i x, where i is he umber of idepede ses of cardialiy i G We show ha he problem of evaluaig he idepedece polyomial of a graph a ay fixed o-zero umber is iracable, eve whe resriced o circulas We provide a formula for he idepedece polyomial of a cerai family of circulas, ad is compleme As a applicaio, we derive a formula for he umber of chords i a -e musical sysem (oe where he raio of frequecies i a semioe is 2 1/ wihou close pich classes 2008 Elsevier BV All righs reserved 1 Iroducio Le G be a graph A subse T of he verex se of G is a idepede se if o wo verices of T are adjace i G We ca ecode he umber of idepede ses of each cardialiy by a geeraig fucio Defiiio 11 ([17] The idepedece polyomial of a graph G o verices is I(G, x = =0 i x, where i is he umber of idepede ses of cardialiy i G By defiiio, he idepedece umber α(g of a graph G is equal o deg(i(g, x, he degree of he idepedece polyomial I(G, x For example, he idepedece polyomial of he 6-cycle C 6 is give by I(C 6, x = 1 + 6x + 9x 2 + 2x 3, as C 6 has i 0 = 1 (he empy se, i 1 = 6, i 2 = 9 (he umber of o-edges of G, ad i 3 = 2 The laer follows as here are precisely wo idepede ses of cardialiy 3, amely {0, 2, 4} ad {1, 3, 5} A variey of graph polyomials, such as chromaic polyomials, machig polyomials, characerisic polyomials, have bee well sudied Idepedece polyomials have bee ivesigaed i a umber of papers [5 8,14,16 22] Oe highly srucured (ad well ow family of graphs are circulas Give 1 ad S Z {0} wih S = { s : s S} = S, he circula C,S of order wih geeraig se S is a graph o V = Z such ha for u, w V, uw is a edge of C,S if ad oly if u w S Such graphs are regular ad verex rasiive, ad arise i a variey of graph applicaios We sudy here he idepedece polyomials of circulas I Secio 2, we show ha for ay 0, he problem of evaluaig he idepedece polyomial I(G, x a x = is iracable, eve whe resriced o circulas I Secio 3, we cosider circulas of he form C d = C,{±1,±2,,±d}, where adjace verices are separaed by a circular disace of a mos d For each pair (, d, we compue a explici formula for he idepedece polyomial I(C d, x as well as I(C d, x, he idepedece polyomial of is compleme I Secio 4, we use Correspodig auhor address: brow@mahsadalca (J Brow X/$ see fro maer 2008 Elsevier BV All righs reserved doi:101016/jdisc

2 J Brow, R Hoshio / Discree Mahemaics 309 ( he heorems from he previous secio o provide a applicaio of idepedece polyomials o music Specifically, we esablish a formula for he umber of chords i a -e musical sysem (whose semioe correspods o a raio of 2 1/ wihou close pich classes I wha follows, for ay polyomial P(x, we shall deoe by [x ]P(x he coefficie of he x erm i P(x I geeral we follow [12] for graph-heoreic ermiology For discussio of releva compuaioal complexiy, we refer he reader o [15,23] 2 The iracabiliy of evaluaig he idepedece polyomial a o-zero umbers Give he idepedece polyomial I(G, x of a graph G, we may be ieresed i evaluaig he polyomial a a paricular poi x = As a example, evaluaig I(G, x a x = 1 gives us he oal umber of idepede ses i he graph Evaluaig a graph polyomial a paricular pois has bee a subjec of much ieres, especially for chromaic polyomials [13,23] I geeral, i is NP-hard o deermie he idepedece polyomial I(G, x, sice we ow ha evaluaig α(g is NP-hard [15] Thus, i is o compuaioally efficie o solve he problem by firs compuig he idepedece polyomial We wish o deermie he complexiy of evaluaig I(G, for a arbirary umber C For = 0 i is obviously polyomial (i is 1, so we oly cosider 0 The equivale problem for chromaic polyomials has already bee solved i [23], where i was show ha evaluaig he chromaic polyomial is #P-hard for all {0, 1, 2} (for hese values, he evaluaio ca be easily be compued i polyomial ime We ow give a complee soluio o he evaluaio problem for idepedece polyomials, eve whe resriced o circulas: we prove ha for ay C {0}, i is #P-hard o evaluae I(G, Furhermore, if = 1, he he problem is #P-complee Firs, we require a defiiio ad a heorem o he lexicographic produc of wo graphs Defiiio 21 For ay wo graphs G ad H, he lexicographic produc is a ew graph G[H] wih verex se V(G V(H such ha ay wo verices (g, h ad (g, h i G[H] are adjace iff (g g or (g = g ad h h The followig heorem shows ha I(G[H], x ca be calculaed direcly from I(G, x ad I(H, x Theorem 22 ([6] For ay graphs G ad H, I(G[H], x = I(G, I(H, x 1 We ca prove our heorem o he compuaioal complexiy of evaluaig I(G, x a x = Theorem 23 Compuig I(G, for a give umber C {0} is #P-hard, eve whe resriced o circula graphs G Proof Suppose here exiss a umber 0 for which every I(G, ca be evaluaed i polyomial-ime, wheever G is a circula I oher words, give ay circula graph G o verices, here exiss a O( algorihm o compue I(G, for some cosa Le G be a fixed circula o verices For each 1 m + 1, defie H m o be he lexicographic produc graph G[K m ] By a heorem i [4], he lexicographic produc of wo circulas is always a circula Sice K m is (rivially a circula, G[K m ] is as well By our assumpio, here is a O(m O( 2 algorihm o compue he value of I(H m, The cosrucio of each H m creaes m 2 + verices ad decides if each pair of verices is adjace i H m The ( umber of pairs of verices i H m is a mos = O( 4, ad so cosrucig each H m ca be doe i polyomial-ime By Theorem 22, I(H m, x = I(G, I(K m, x 1 = I(G, mx Therefore, I(H m, = I(G, m for all 1 m + 1 We ow ha here is a O(m O( 2 algorihm o compue he value of I(H m,, for each m Therefore, i aes O( 2+1 seps o evaluae I(G, x for each x = m Sice 0, hese values of x are disic We ow ha he idepedece polyomial of G is I(G, x = i 0 + i 1 x + i 2 x i x, for some iegers i (Noe ha deg(i(g, x G = Leig x = m for each 1 m + 1, we have a sysem of + 1 equaios ad + 1 uows i 0 + i 1 + i i = I(G, i 0 + i 1 (2 + i 2 ( i (2 = I(G, 2 i 0 + i 1 ( i 2 (( i (( + 1 = I(G, ( + 1 This sysem has a uique soluio iff he marix (2 2 (2 M = 1 ( + 1 (( (( + 1

3 2294 J Brow, R Hoshio / Discree Mahemaics 309 ( Fig 1 The cycle power graph C 2 9 has a o-zero deermia M is a Vadermode marix, ad he formula for is deermia is de(m = ( +1 2 =1! Sice de(m 0, his sysem has a uique soluio (i 0, i 1,, i This sysem of equaios ca be solved i O( 3 ime usig Gaussia elimiaio, ad so each of he i s ca be deermied i polyomial ime, which i ur, gives us he idepedece polyomial I(G, x For ay graph G, we have foud a O( 4 + O( O( 3 algorihm o deermie he formula for I(G, x Sice deg(i(g, x = α(g, we have show ha α(g ca be compued i polyomial-ime for ay circula G This coradics he resul [11] ha o such algorihm exiss Sice i is NP-hard o evaluae α(g for a arbirary circula G [11], we coclude ha i is #P-hard o evaluae I(G,, for all o-zero C We oe ha whe = 1, i is #P-complee o evaluae I(G, This follows sice I(G, 1 simply represes he oal umber of idepede ses i G, ad idepede ses are recogizable i polyomial ime Thus, for he specific case = 1, he problem of evaluaig I(G, is #P-complee I ligh of Theorem 23, i is of ieres o fid families of graphs (ad ideed families of circulas for which we ca fid explici formulas for heir idepedece polyomials 3 The idepedece polyomials of a family of circulas Le d 1 be a fixed ieger I he circula C d, wo verices are adjace iff heir circular disace is a mos d, ha is uw is a edge iff u w d The graph C d is also ow as he dh power of he cycle C Powers of cycles have bee a rich sudy of ivesigaio [2,3,24 26], wih impora coecios o he aalysis of perfec graphs [1,9,10,27] The graph C 2 9 is illusraed i Fig 1 For oaioal coveiece, we adop he followig defiiio, sice d 1 will always be fixed Defiiio 31 Le d 1 be a fixed ieger For each, se A = C d I his secio, we deermie a formula for I(A, x, for all 1 By defiiio, oe ha A is he complee graph K for 2d + 1, ie, I(A, x = I(K, x = 1 + x Thus, we may assume ha 2d + 2 Lemma 32 I(A, x = I(A 1, x + x I(A d 1, x, for all 2d + 2 Proof Sice 2d + 2, we have α(a 2 We see rivially ha he x 0 ad x 1 coefficies are equal i he give ideiy So fix 2 We will show ha he x coefficies are equal as well Le {v 1, v 2,, v } be a idepede se of cardialiy 2 i A, wih 0 v 1 < v 2 < < v 1 Sice he circular disace exceeds d for all o-adjace verices u ad v i A, we have v i+1 v i > d for all 1 i 1, ad + (v 1 v > d This ca be see by placig pois equally aroud a circle, ad oicig ha each (adjace pair of chose verices is separaed by disace greaer ha d We classify our idepede ses {v 1, v 2,, v } of A io wo families: (a S 1 = {{v 1, v 2,, v } idepede i A : v v 1 = d + 1} (b S 2 = {{v 1, v 2,, v } idepede i A : v v 1 > d + 1}

4 J Brow, R Hoshio / Discree Mahemaics 309 ( Sice S 1 S 2 =, i follows ha [x ]A = S 1 + S 2 We will show ha S 1 = [x 1 ]A d 1 ad S 2 = [x ]A 1 Case 1: Provig S 1 = [x 1 ]A d 1 We esablish a bijecio φ bewee S 1 ad he se of ( 1-uples ha are idepede i A d 1 This will prove ha S 1 = [x 1 ]A d 1 For each eleme of S 1, defie φ(v 1, v 2,, v = {v 1, v 2,, v 1 } Sice v = v 1 + (d + 1, φ is oe-o-oe Cosruc he graph A by coracig all of he verices from he se {v 1 + 1, v 1 + 2,, v } o v 1 The A A d 1 We claim ha φ(v 1, v 2,, v is a idepede se of A iff {v 1, v 2,, v } is a eleme of S 1 Noe ha φ(v 1, v 2,, v is a idepede se of A iff (a v i+1 v i > d for 1 i 2 (b ( d 1 + v 1 v 1 > d Also, {v 1, v 2,, v } is a eleme of S 1 iff (a v i+1 v i > d for 1 i 2 (b v v 1 = d + 1 (c + v 1 v > d We ow show ha hese wo ses of codiios are equivale Noe ha he codiio v i+1 v i > d for 1 i 2 is rue i boh cases If φ(v 1, v 2,, v = {v 1, v 2,, v 1 } is a idepede se of A, he ( d 1 + v 1 v 1 > d Le v = v 1 + (d + 1 The, {v 1, v 2,, v 1, v } is a idepede se of A, sice ( d 1 + v 1 (v (d + 1 > d, or + v 1 v > d Therefore, {v 1, v 2,, v } is a eleme of S 1 Now we prove he coverse If {v 1, v 2,, v } is a eleme of S 1, he v v 1 = d + 1 ad + v 1 v > d Addig, his implies ha (v v 1 + ( + v 1 v > 2d + 1, or ( d 1 + v 1 v 1 > d Hece, φ(v 1, v 2,, v is a idepede se of A Therefore, we have esablished ha φ is a bijecio bewee he ses i S 1 ad he idepede ses of cardialiy 1 i A A d 1 We coclude ha S 1 = [x 1 ]A d 1 Case 2: Provig S 2 = [x ]A 1 We ow esablish a bijecio ϕ bewee S 2 ad he se of idepede -uples i A 1 For each eleme (v 1, v 2,, v 1, v of S 2, defie ϕ(v 1, v 2,, v 1, v = {v 1, v 2,, v 1, v 1} Observe ha ϕ is oe-o-oe Cosruc he graph A by coracig v o v 1 The, A A 1 We claim ha ϕ(v 1, v 2,, v is a idepede se of A iff {v 1, v 2,, v } is a eleme of S 2 Noe ha ϕ(v 1, v 2,, v is a idepede se of A iff (a v i+1 v i > d for 1 i 2 (b (v 1 v 1 > d (c ( 1 + v 1 (v 1 > d Also, {v 1, v 2,, v } is a eleme of S 2 iff (a v i+1 v i > d for 1 i 2 (b v v 1 > d + 1 (c + v 1 v > d Clearly, hese ses of codiios are equivale Therefore, we have esablished ha ϕ is a bijecio bewee he ses i S 2 ad he idepede ses of cardialiy i A A 1 We coclude ha S 2 = [x ]A 1 Therefore, we have show ha [x ]A = [x 1 ]A d 1 + [x ]A 1, which implies ha I(A, x = I(A 1, x + x I(A d 1, x Now we fid a explici formula for I(A, x Theorem 33 Le d + 1 The, d+1 I(A, x = I(C d, x = =0 ( d x d

5 2296 J Brow, R Hoshio / Discree Mahemaics 309 ( Proof By Lemma 32, I(A, x = I(A 1, x + x I(A d 1, x, for 2d + 2 We will prove he heorem usig geeraig fucios { I(A, x for d + 1 Le f = 1 for 1 d d + 1 for = 0 Each f is a polyomial i x Firs, we verify ha f = f 1 +xf d 1, for all d+1 This recurrece is rue for 2d+2, by Lemma 32 For d + 2 2d + 1, we have f = 1 + x = (1 + ( 1x + x 1 = f 1 + xf d 1 Fially, for = d + 1, we have f d+1 = 1 + (d + 1x = f d + xf 0 Thus, f = f 1 + xf d 1, for all d + 1 Le F(x, y = p=0 f py p For each d + 1, we will show ha ( ( ( d d 1 d [x y ]F(x, y = + d 1 Sice f = f 1 + xf d 1, for all d + 1, we have f y = f 1 y + f d 1 xy =d+1 F(x, y =d+1 =d+1 = d ( d d 1 f y = y F(x, y f y + xy d+1 F(x, y =0 =0 F(x, y(1 y xy d+1 = f 0 + f 1 y + F(x, y(1 y xy d+1 = f 0 + f 1 y + d d 1 f y f 0 y f y +1 =2 d y f 0 y =2 F(x, y(1 y xy d+1 = (d y (d + 1y F(x, y = (d + 1 dy(1 y xy d+1 1 F(x, y = (d + 1 dy (y + xy d+1 F(x, y = (d + 1 dy =0 y (1 + xy d =0 =1 d =2 ( F(x, y = (d + 1 dy x u y +du u =0 u=0 ( ( F(x, y = (d + 1 x u y +du d x u y +du+1 u u,u=0,u=0 Now we exrac he x y coefficie of F(x, y The las lie will follow from Pascal s Ideiy ( ( [x y ]F(x, y = [x y ](d + 1 x u y +du [x y ]d x u y +du+1 u u,u=0,u=0 ( ( d d 1 = (d + 1 d ( [( ( ] d d d 1 = + d ( ( d d 1 = + d 1 ( d = d Therefore, we have prove ha [x ]I(A, x = [x y ]F(x, y = d ( d y

6 J Brow, R Hoshio / Discree Mahemaics 309 ( We oe ha his coefficie is o-zero precisely whe d, which is equivale o he codiio deg(i(a, x = d+1 We coclude ha I(C d, x = I(A, x = ( d+1 d =0 x d We remar ha our formula for he special case d = 1, amely 2 ( I(C, x = x, =0 d+1 Hece, has previously appeared i he lieraure [17], via a alerae mehod of proof Now we compue a formula for I(C d, x, he idepedece polyomial of he compleme of C d Sice d will remai fixed, we iroduce he followig defiiio for oaioal coveiece Defiiio 34 Le d 0 be a fixed ieger For each 2d+2, defie he graph B o be he compleme of A Specifically, B = A = C d Noe ha if = 2d + 2, he B is he disjoi uio of d + 1 copies of K 2, ad so I(B, x = [I(K 2, x] d+1 = (1 + 2x d+1 If = 2d + 3, he B = B 2d+3 is simply he cycle C 2d+3, ad a formula for his idepedece polyomial was esablished i Theorem 33 Fially, if 3d + 1, he he formula for he idepedece polyomial is sraighforward o prove The correc formula was firs esablished i a paper by Michael ad Traves Proposiio 35 ([28] Le 3d + 1 The, I(B, x = 1 + x(1 + x d Therefore, we are lef wih he case 2d + 4 3d As we will see, deermiig a formula for I(B, x = I(A, x i his case will be exremely complicaed, ad he proof will require may echical lemmas Firs, we iroduce he followig defiiio Defiiio 36 For each -uple {v 1, v 2,, v } of he verices of a graph G o verices, wih 0 v 1 < v 2 < < v 1, he differece sequece is (d 1, d 2,, d = (v 2 v 1, v 3 v 2,, v v 1, + v 1 v Differece sequeces will be of remedous help i couig he umber of idepede ses We will carefully sudy he srucure of hese differece sequeces, ad deermie a direc correlaio o idepede ses As we did i he proof of Lemma 32, we ca visualize differece sequeces as follows: spread verices aroud a circle, ad highligh he chose verices v 1, v 2,, v Now, le d i be he disace bewee v i ad v i+1, for each 1 i (oe: v +1 := v 1 I oher words, he d i s jus represe he disaces bewee each pair of highlighed verices By his reasoig, i is clear ha i=1 d i = ad ha v j = v 1 + j 1 i=1 d i for each 1 j Isead of direcly eumeraig he idepede ses I of B, i will be easier o deermie all possible differece sequeces D ha correspod o a idepede se of B, ad he eumerae he umber of idepede ses correspodig o hese differece sequeces For oaioal coveiece, we iroduce he followig defiiio Defiiio 37 A differece sequece D = (d 1, d 2,, d of he circula B is valid if o cyclic subsequece of cosecuive d i s sum o a eleme i {d + 1, d + 2,, 2 } By a cyclic subsequece of cosecuive erms, we refer o subsequeces such as {d 2, d 1, d, d 1, d 2, d 3, d 4 } From ow o, whe we refer o subsequeces of D, his will auomaically iclude all cyclic subsequeces We oe ha each idepede se I of B maps o a valid differece sequece D The followig lemma is immediae from he defiiios, ad so we omi he proof Lemma 38 Le he se I = {v 1, v 2,, v } have he differece sequece D = (d 1, d 2,, d The, I is idepede i B iff D is valid We will ow describe a explici cosrucio of all valid differece sequeces wih elemes, ad his will yield he oal umber of idepede ses wih cardialiy The desired formula for I(B, x is he followig Theorem 39 Le (, d be a ordered pair wih 2d + 2 Le B = A = C d, ad se r = 2d 2 0 The, I(B, x = 1 + d r+2 l=0 ( d lr x +1 (1 + x d l(r I is easy o show ha Proposiio 35 follows immediaely from Theorem 39 We omi he deails To prove Theorem 39, we require several echical combiaorial lemmas

7 2298 J Brow, R Hoshio / Discree Mahemaics 309 ( I he ex lemma, we cou he umber of m-uples (Q 1, Q 2,, Q m wih a fixed sum ha coai a oal of o-zero elemes amog he Q i s I his case, each Q i is a (possibly empy sequece of posiive iegers ( Lemma 310 Le a 1, a 2,, a m be o-egaive iegers wih sum The here are exacly m-uples (Q 1, Q 2,, Q m ha coai a oal of o-zero elemes amog he Q i s, where each Q i is a (possibly empy sequece of posiive iegers whose sum is a mos a i Proof Wrie dow a srig of oes, ad place m 1 bars i bewee he oes o creae he pariio correspodig o he m-uple (a 1, a 2,, a m Now selec ay of he oes As a example, we demosrae his for he case (a 1, a 2, a 3 = (5, 6, 4, m = 3, = 15, ad = 6 1, 1, 1, 1, 1 1, 1, 1, 1, 1, 1 1, 1, 1, 1 ( Clearly, here are ways o selec exacly oes from his srig We map each selecio o a uique m-uple (Q 1, Q 2,, Q m which coais a oal of o-zero elemes amog he Q i s, so ha he sum of he elemes i each Q i is a mos a i Cosider he subsrig of a i oes i he ih pariio If o elemes are seleced from his subsrig, se Q i = Oherwise, le he seleced elemes i he ih pariio be i posiios r 1, r 2,, r p, where 1 r 1 < r 2 < < r p a i Now defie Q i = (r 2 r 1, r 3 r 2,, r p r p 1, a i + 1 r p I oher words, each Q i ca be hough of as he differece sequece of he p chose verices i a circula of order a i I he above example, our selecio of he s correspods o he ses Q 1 = {2, 2}, Q 2 = {1, 4, 1}, Q 3 = {3}, which coai ( a oal of = 6 o-zero elemes Noe ha for each i, Q i = a i + 1 r 1 a i This cosrucio guaraees ha each of he selecios maps o a uique m-uple (Q 1, Q 2,, Q m wih a oal of o-zero elemes, so ha Q i a i Give such a m-uple, we ow jusify ha we ca deermie he uique way he oes were seleced from he srig For each subsrig of oes i he ih pariio, we are give Q i From he above defiiio for Q i, we ca deermie he values (or posiios of he r j s by sarig a r p ad calculaig bacwards From r p, we ca uiquely compue r p 1, r p 2, ad so o, uil we have deermied all of he r j s Sice we ca do his for each i, each selecio of he m-uple (Q 1, Q 2,, Q m correspods o a uique selecio of elemes from a srig of oes Hece, his cosrucio is bijecive, ad our proof is complee We ow iroduce l-cosrucible differece sequeces While he defiiio may appear corived, i is precisely he isigh we eed o cou he umber of idepede ses of B We will show ha every differece sequece is uiquely l-cosrucible, for exacly oe ieger l 0 The i our proof of Theorem 39, we will eumerae he umber of l- cosrucible differece sequeces o deermie he umber of idepede ses of each cardialiy Defiiio 311 Le D be a differece sequece of B = A = C d, where 2d + 2 The, for each ieger l 0, D is l-cosrucible if D ca be expressed i he form D = Q 1, p 1, Q 2, p 2,, Q +1, p +1 such ha he followig properies hold 1 Each p i is a ieger saisfyig p i 2d 2 Each Q i is a sequece of iegers, possibly empy 3 Le S be ay (cyclic subsequece of cosecuive erms i D wih sum S If S coais a mos l of he p i s, he S d Oherwise, S d We ow prove ha every valid differece sequece ca be expressed uiquely as a l-cosrucible sequece, for exacly oe l 0 We will he eumerae he umber of l-cosrucible sequeces for each l, which will give us he oal umber of valid differece sequeces A differece sequece D of B = A = C d is valid iff o subsequece of cosecuive erms adds up o a eleme i {d + 1, d + 2,, } Sice he compleme of ay cosecuive subsequece of D is also a cosecuive subsequece of 2 D, here exiss a cosecuive subsequece wih sum iff here exiss a cosecuive subsequece wih sum I oher words, D is valid iff o subsequece of cosecuive erms sums o a eleme i [d + 1, d 1] By he hird propery i he defiiio of l-cosrucibiliy (see above, every l-cosrucible sequece is ecessarily valid because every subsequece of cosecuive erms has sum a mos d or a leas d, ad hece falls ouside of he forbidde rage [d + 1, d 1] So every l-cosrucible sequece is a valid differece sequece I he ex wo lemmas, we prove ha every valid differece sequece is uiquely l-cosrucible, for exacly oe l 0 Firs, we cosruc a l ha saisfies he codiios, ad he we prove ha o oher l suffices To suppleme he echical deails of he followig proof, le us describe our mehod by illusraig a example Cosider he case = 89 ad d = 40 I is sraighforward o show ha he differece sequece D = {9, 1, 9, 1, 9, 20, 10, 19, 2, 9} is valid, ie, o subsequece of cosecuive elemes sums o ay S [41, 49] We prove ha his differece sequece D is uiquely 2-cosrucible, up o cyclic permuaio

8 J Brow, R Hoshio / Discree Mahemaics 309 ( Lemma 312 Le D be a valid differece sequece of B The here exiss a ieger l 0 such ha D is l-cosrucible For his ieger l, D is l-cosrucible i a uique way up o cyclic permuaio, ie, here is oly oe way o selec he Q i s ad p i s so ha D is l-cosrucible Proof Le D = R 1 1 R 2 2 R m m, where each i 2d ad each R i is a (possibly empy sequece of erms, all of which are less ha 2d Thus, each D has a uique represeaio i his form, up o cyclic permuaio I our example, 2d = 9 Wihou loss, assume 1 = 20 I his case, we mus have R 2 =, 2 = 10, R 3 =, 3 = 19, R 4 = {2}, 4 = 9, R 5 =, 5 = 9, R 6 = {1}, 6 = 9, R 7 = {1}, 7 = 9, ad R 1 = I oher words, we have D = 20, 10, 19, 2, 9, 9, 1, 9, 1, R R 6 6 R 7 Le l 0 be he larges ieger such ha for ay subsequece X of cosecuive erms of D, X d if X icludes a mos l of he i s I our example, l < 3 sice X = {20, 10, 19} icludes hree of he i s, ad X = 49 > d By ispecio, i ca be checed ha l = 2 For his l 0, we prove ha D is l-cosrucible, ad ha he assigme of Q i s ad p i s is uique, up o cyclic permuaio Firs suppose ha m Noe ha R R R l + l d sice his series coais exacly l of he i s Similarly, R l+1 + l R + d If m, he = D 2d <, a coradicio Thus, m + 1 If m = + 1, he we ca se Q i = R i ad p i = i for each i The each D is l-cosrucible, sice D d if D coais a mos l of he p i s, ad D d oherwise I is clear ha his is he oly assigme ha eables D o be l-cosrucible, up o cyclic permuaio So suppose ha m > + 1 I his case, we will assig he p i s ad Q i s from he se of i s ad R i s All of he p i s will be chose from he se of i s, while all of he Q i s will be deermied from he R i s, as well as ay lefover i s o icluded amog he p i s By he defiiio of he idex l 0, here mus be a subsequece X coaiig l + 1 of he i s such ha is sum exceeds d Sice D is valid, o subsequece of cosecuive erms ca sum o ay umber i [d + 1, d 1] Therefore, X > d implies ha X d Cyclically permue he elemes of D so ha his subsequece X appears a he fro of D, ie, redefie he R i s ad i s so ha we have 1 + R R l+1 + l+1 d The se p i = i for 1 i l + 1 ad Q i = R i for 2 i l + 1 I our example, we have X = {20, 10, 19}, p 1 = 20, Q 2 =, p 2 = 10, Q 3 =, ad p 3 = 19 Noe ha his assigme of p i s ad Q i s is ecessary for D o be l-cosrucible: if ay of hese Q i s coais a j erm, he we will obai a coradicio because he above subsequece X will have a mos l of he p i s, bu is sum will exceed d If D is l-cosrucible, we require he chose p i s ad Q i s o saisfy Q 2 + p Q l+1 + p l+1 + Q l+2 d, 7 sice his subsequece coais l of he p i s Also, we require Q 2 + p Q l+1 + p l+1 + Q l+2 + p l+2 d, sice his subsequece coais l + 1 of he p i s Le T = Q 2 + p Q l+1 + p l+1 The Q l+2 d T ad Q l+2 + p l+2 d T Sice each p i ad Q i has already bee assiged for 2 i l + 1, T is a fixed ieger From hese wo iequaliies, we claim ha Q l+2 is uiquely deermied Noe ha for some 0, Q l+2 mus be he firs elemes of he sequece X = R l+2, l+2, R l+3, l+3, R m, m, R 1 Furhermore, p l+2 would have o be he ex erm, ie, he ( + 1h erm of X We claim ha mus be he larges ieger such ha he firs erms of X sum o a mos d T This choice is uique because if were o he larges ieger, he Q l+2 + p l+2 d T, ad ha coradics he iequaliy Ql+2 +p l+2 d T Sice is uiquely deermied, Q l+2 mus represe he firs elemes of X, i order for D o be l- cosrucible Furhermore, p l+2 mus be he ex erm i his subsequece I our example, T = 29, X = {2, 9, 9, 1, 9, 1, 9}, Q 4 = {2, 9}, ad p 4 = 9 Cosider his sum T + Q l+2 + p l+2 > d By our choice of, his sum exceeds d Sice D is valid, his sum mus be a leas d, sice his oal represes he sum of a subsequece of cosecuive erms i D Therefore, he admissibiliy of D implies ha Q l+2 + p l+2 d T Hece, by our cosrucio, oce we fix p i ad Q i for 2 i l + 1, he Q l+2 ad p l+2 are uiquely deermied, ad saisfy he properies of l-cosrucibiliy Noe ha p l+2 mus saisfy he iequaliy p l+2 2d sice T + Q l+2 d ad T + Q l+2 + p l+2 d By he same argume, each p i 2d This proves ha each p i is chose from he se of i s Similarly, Q i ad p i are uiquely deermied for i = l + 2, i = l + 3, ad all he way up o i = + 1 Oce Q +1 ad p +1 are chose, we are lef wih useleced erms for some 0 The our oly choice is o assig hese erms o Q 1 Thus, his assigme of p i s ad Q i s mus be uique, up o cyclic permuaio This complees he proof

9 2300 J Brow, R Hoshio / Discree Mahemaics 309 ( I our example wih (, d = (89, 40, we have already deermied p i ad Q i for each 1 i 4 By applyig he above mehod, we see ha Q 5 = {1}, p 5 = 9, ad Q 1 = {1, 9} We ca readily verify ha his represeaio of D io p i s ad Q i s saisfies he properies of a l-cosrucible sequece Thus, we have show ha every 2-cosrucible represeaio of D mus be a cyclic permuaio of 1, 9, 20, 10, 19, 2, 9, 9, 1, 9 Q p 1 1 p 2 p 3 Q p 4 4 Q 5 p 5 The ex lemma shows ha D is l-cosrucible for oly oe l 0 Lemma 313 If D is l-cosrucible, he D is o l -cosrucible, for ay l l Proof Suppose ha D is boh l-cosrucible ad l -cosrucible Wihou loss, suppose l < l Sice D is l-cosrucible, we ow ha D ca be expressed as D = Q 1, p 1, Q 2, p 2,, Q +1, p +1, such ha S d if S coais a mos l of he p i s, ad S d oherwise If D is l -cosrucible, he D ca also be expressed as D = Q 1, p 1, Q 2, p 2,, Q, p, such ha S d if S coais a mos l of he p i s, ad S d oherwise For each 1 j + 1, defie X j o be he subsequece X j = p j, Q j+1, p j+1,, Q j+l, p j+l, where he idices are reduced mod ( + 1 Sice X j coais exacly l + 1 of he p i s, X j d This sequece X j appears exacly as a subsequece of cosecuive erms i D = Q 1, p 1, Q 2, p 2,, Q +1, p +1 Sice X j d, i follows ha X j mus coai a leas (l + 1 of he p i s, sice D is l-cosrucible For each 1 j + 1, defie Γ (Q j o be he umber of p i s ha appear i Q j, ad defie Γ (p = j 1 if p j = p i for some i, ad Γ (p = j 0 oherwise Sice X j coais a leas l + 1 of he p i s, we mus have Γ (p + Γ j (Q + Γ j+1 (p + + Γ j+1 (Q j+l + Γ (p j+l l + 1 Summig over all 1 j + 1, we have l Γ (Q + j (l + 1 Γ (p j (l + 1( + 1 This ideiy follows because each Γ (Q j is coued l imes ad each Γ (p j is coued l + 1 imes This iequaliy ca be rewrie as: Γ (Q j (l + 1( + 1 (l Γ (p j l For each 1 j + 1, defie Y j o be he subsequece Y j = Q j, p j, Q j+1, p j+1,, Q j+l 1, p j+l 1, Q j+l, where he idices are reduced mod ( + 1 Sice Y j coais exacly l of he p i s, Y j d This sequece Y j appears exacly as a subsequece of cosecuive erms i D = Q 1, p 1, Q 2, p 2,, Q +1, p +1 Sice Y j d, i follows ha Y j coais a mos l of he p i s, sice D is l-cosrucible Therefore, we have Γ (Q + Γ j (p + Γ j (Q + Γ j+1 (p + + Γ j+1 (p j+l + Γ 1 (Q j+l l Summig over all 1 j + 1, we have (l + 1 Γ (Q + j l Γ (p j l( + 1

10 J Brow, R Hoshio / Discree Mahemaics 309 ( This iequaliy ca be rewrie as Γ (Q j l( + 1 l +1 l + 1 Γ (p j So ow we have wo iequaliies i erms of Γ (Q j ad Γ (p j From hese wo iequaliies, we have (l + 1( + 1 (l + 1 Γ (p j l( + 1 l Γ (p j l l + 1 (l + 1(l + 1( + 1 (l Γ (p j ll ( + 1 l 2 Γ (p j ( + 1(l + 1(l + 1 ( + 1ll ( + 1 Γ (p j Γ (p j ll + l + l + 1 ll Γ (p j = l + l + 1 Γ (p j > + 1 (sice l > l By he Pigeohole Priciple, we mus have Γ (p > j 1 for some idex j However, each Γ (p j 1 ad his gives us our desired coradicio Therefore, we have show ha for ay l l, D is o l -cosrucible if D is l-cosrucible We are fially ready o prove Theorem 39 Proof By he defiiio of a l-cosrucible sequece, every subsequece of cosecuive erms has a sum ouside he rage [d + 1, d 1] Therefore, each l-cosrucible sequece is valid i B, for every l 0 By Lemmas 312 ad 313, we have show ha here is a bijecio bewee he se of valid differece sequeces of B ad he uio of all l-cosrucible sequeces for l 0 Every valid differece sequece D correspods o a uique l-cosrucible sequece, for exacly oe l 0 To deermie he umber of valid differece sequeces of B, i suffices o deermie he umber of l-cosrucible sequeces for each l 0, ad he eumerae is uio Le D be a l-cosrucible sequece, for some fixed l 0 Thus, D is valid i B By defiiio, ay subsequece of cosecuive erms coaiig l of he p i s mus sum o a mos d Cosider a l-cosrucible sequece D = Q 1, p 1, Q 2, p 2,, Q +1, p +1 We eumerae he umber of all possible l- cosrucible sequeces, for his fixed l 0 We will show ha each l-cosrucible sequece D mus be geeraed i he followig way: (a Choose (a 1, a 2,, a +1 o be a ordered ( + 1-uple of o-egaive iegers wih sum = ( + 1d l (b Selec Q 1, Q 2,, Q +1 so ha Q j a j+l+1 for each 1 j + 1 Noe ha for j l + 1, he idex j + l + 1 is reduced mod ( + 1 (c From his, each p j is uiquely deermied, ad saisfies p j 2d (d The sequece D = Q 1, p 1, Q 2, p 2,, Q +1, p +1 is l-cosrucible Each of hese seps is easy o eumerae, ad his will eable us o cou he oal umber of l-cosrucible differece sequeces Defie X j = Q j, p j,, Q j+l 1, p j+l 1 for each 1 j + 1, where he idices are reduced mod ( + 1 Sice X j coais l of he p i s, X j d Le a j be he ieger for which X j = d a j The each a j 0 Le X j = X j, Q l+j The X j d because X j coais oly l of he p i s Hece, X j = X j + Q l+j d, which implies ha Q l+j a j This is rue for each j, so addig l + 1 o boh idices ad reducig mod ( + 1, we have Q j a j+l+1 Noe ha Q j + p j = X j+1 X j+l+1 = (d a j+1 (d a j+l+1 = 2d + a j+1 + a j+l+1

11 2302 J Brow, R Hoshio / Discree Mahemaics 309 ( Sice Q j a j+l+1, i follows ha p j 2d+a j+1 2d, which is cosise wih he defiiio of l-cosrucibiliy Le = a j We have X j = d a j for each j Addig hese + 1 sums, we have l = ( + 1d, or = ( + 1d l 0 So is fixed ( Sice he a j s are o-egaive iegers wih sum, a well-ow combiaorial ideiy shows ha here are ways o selec he (+1-uple (a 1, a 2,, a +1 For each of hese (+1-uples, we selec our Q j s so ha Q j a j+l+1 for each 1 j + 1 By Lemma ( 310, if our Q j s have a oal of o-zero elemes amog hem, he our selecio of he Q j s ca be made i exacly ways This l-cosrucible sequece D will coai a oal ( of ( erms, wih of hem comig from he uio of he Q j s, ad oe for each of he + 1 p i s So here are possible l-cosrucible sequeces wih erms + Therefore, here are his may valid differece sequeces of B wih erms Noe ha some of hese sequeces are cyclic permuaios of ohers, ad we will ae his io accou whe we deermie he umber of idepede ses wih verices ( ( + Le Ψ be he se of pairs (v, D, where v is a verex of B ad D is ay of he l-cosrucible sequeces wih ( ( elemes Each of he pairs i Ψ will correspod o a idepede se I wih verices: + I = {v, v + d 1, v + d 1 + d 2,, v + d 1 + d d + }, where he elemes are reduced mod ad arraged i icreasig order We ow jusify ha each idepede se I appears exacly ( + 1 imes by his cosrucio The ey isigh is ha each D is a l-cosrucible sequece, ad hece has he followig form: D = Q 1, p 1, Q 2, p 2,, Q +1, p +1 Therefore, here are exacly ( + 1 cyclic permuaios of D so ha i reais he form of a l-cosrucible sequece: for each cyclic permuaio, he sequece begis wih Q i, for some 1 i + 1 Thus, we mus divide he oal umber of idepede ses by ( + 1, as each oe is repeaed his may imes I oher words, here are +1 ( + ( idepede ses wih verices Sice his is rue for each l 0 ad 0 = ( + 1d l, i follows ha I(B, x = 1 + ( ( + x l 0 =0 = 1 + ( + ( x +1 x + 1 l 0 =0 = 1 + ( + x +1 (1 + x + 1 l 0 = 1 + ( ( + 1d l( 2 x +1 (1 + x (+1d l + 1 l 0 Noe ha we require = ( + 1d l 0 for here o be ay idepede ses Thus, l d Leig r = 2d 2, 2d we coclude ha I(C,{±d+1,±d+2,,± 2 }, x = 1 + d r+2 l=0 This cocludes he proof of Theorem 39 ( d lr x +1 (1 + x d l(r A applicaio o music The 12-semioe music scale cosiss of he pich classes C, C #, D, D #, E, F, F #, G, G #, A, A #, ad B Each oe is ideified wih is pich class (ie, each C refers o he same oe, regardless of is ocave These pich classes are he musical aalogue of equivalece classes Suppose we wa o play a chord cosisig of 3 differe pich classes from his scale Clearly, he umber of differe ( possibiliies is 12 Bu if we were o iroduce forbidde iervals ad as for he umber of chords we could play wih his resricio, he we ca aswer his problem usig idepedece polyomials I paricular, if he forbidde iervals

12 J Brow, R Hoshio / Discree Mahemaics 309 ( correspod o pich classes ha are close ogeher (ad hece, dissoa, we show ha his problem ca be aswered from he idepedece polyomial I(C d, x As a simple example, suppose ha we are forbidde o iclude ay chord wih wo pich classes separaed by a semioe or oe (for example, C ad C #, or G ad A I oher words, if we were o draw a graph wih hese 12 pich classes as our verices, we would require every pair of pich classes o be separaed by a disace of a leas hree (ie, a mior hird, o avoid ay semioes or oes Now we ca as how may possible chords ca be played wih his give resricio I(C 2 12, x Mahemaically, his is equivale o he problem of evaluaig he idepedece polyomial I(C 2 12, x, ad he subsiuig x = 1 o deermie our aswer I oher words, every possible chord is some idepede se of size a leas 3 i he circula C 2 12, sice each pair of pich classes i a idepede se is separaed by a leas a mior hird (hree semioes By Theorem 33, 4 ( I(C 2, x = x = x + 42x x 3 + 3x =0 Thus, I(C 2 12, 1 = 98 We coclude ha here are 98 ( = 43 possible chords ha ca be played, excludig he 55 rivial chords of less ha hree pich classes (correspodig o he 55 idepede ses of size a mos 2 i C 2 12 We ca geeralize he 12-semioe ocave o he -semioe ocave As i he 12-semioe ocave, he -semioe ocave is divided io equally empered oes, each formed by muliplyig he frequecy by 2 1 Musicias refer o his as he - e scale (where e is a acroym of Toe Equal-Tempered; see, for example, [29] Tradiioal Thai isrumes are ued o a scale ha is approximaely 7-e, ad various composers have wrie music i -e scales for oher values of (oe commo oe, 19-e, is well-suied as he raio of 3/2, a perfec fifh, ca be approximaed very closely I a -e scale he raio bewee ay wo semioes is cosa Sice oes wih close frequecies soud dissoa whe played ogeher, we ca require ha o chord iclude wo pich classes separaed by d semioes or less, for some ieger d 1 Le f (, d be he umber of possible o-rivial chords ha ca be played wih his resricio By Theorem 33, he aswer is simply f (, d = d+1 =3 d ( d, which we derive from evaluaig I(C d, x a x = 1, ad he subracig he umber of rivial chords wih less ha hree pich classes This gives us he formula for he umber of opimal chords We could also deermie he umber of leas-opimal chords, where each pair of pich classes is separaed by a mos d semioes I oher words, each pair of oes soud dissoa whe played ogeher, ie, he selecio of he pich classes is he wors possible Le g(, d be he umber of possible o-rivial chords ha ca be played wih his reverse resricio, where r = 2d 2 0 By Theorem 39, he aswer is simply g(, d = 1 + d r+2 =3 ( d r 2 d (r We remar ha our couig mehod also exeds o equally empered subdivisios of sreched ad shrue ocaves, ha is, hose scales where he raio of frequecy of he op ad boom frequecies is larger or smaller ha 2, respecively Refereces [1] C Berge, Moivaios ad hisory of some of my cojecures, Discree Mahemaics 165 ( [2] J-C Bermod, C Peyra, Iduced subgraphs of he power of a cycle, SIAM Joural o Discree Mahemaics 2 ( [3] JA Body, SC Loce, Triagle-free subgraphs of powers of cycles, Graphs ad Combiaorics 8 ( [4] I Broere, JH Haigh, Producs of circula graphs, Quaesioes Mahemaica 13 ( [5] JI Brow, K Dilcher, RJ Nowaowsi, Roos of idepedece polyomials of well covered graphs, Joural of Algebraic Combiaorics 11 ( [6] JI Brow, CA Hicma, RJ Nowaowsi, O he locaio of roos of idepedece polyomials, Joural of Algebraic Combiaorics 19 ( [7] JI Brow, CA Hicma, RJ Nowaowsi, The idepedece fracal of a graph, Joural of Combiaorial Theory Series B 87 ( [8] JI Brow, RJ Nowaowsi, Boudig he roos of idepedece polyomials, Ars Combiaoria 58 ( [9] M Chudovsy, N Roberso, P Seymour, R Thomas, The srog perfec graph heorem, Aals of Mahemaics 164 ( [10] V Chváal, O he srog perfec graph cojecure, Joural of Combiaorial Theory Series B 20 ( [11] B Codeoi, I Gerace, S Viga, Hardess resuls ad specral echiques for combiaorial problems o circula graphs, IEEE Trasacios o Compuers 48 ( [12] R Diesel, Graph Theory, Spriger-Verlag, New Yor, 2000 [13] K Dohme, A Pöiz, P Tima, A ew wo-variable geeralizaio of he chromaic polyomial, Discree Mahemaics ad Theoreical Compuer Sciece 6 ( [14] DC Fisher, AE Solow, Depedece polyomials, Discree Mahemaics 82 (

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