A Lower Bound for the Rectilinear Crossing Number.

Transcription

1 A Lower Boud for the Rectiliear Crossig Number. Berardo M. Ábrego, Silvia Ferádez-Merchat Califoria State Uiversity Northridge ver 4 Abstract We give a ew lower boud for the rectiliear crossig umber cr() of the complete geometric graph K.Weprovethatcr() 3 4 adweextedtheproofof the result to pseudoliear drawigs of K. Itroductio The crossig umber cr (G) of a simple graph G is the miimum umber of edge crossigs i ay drawig of G i the plae, where each edge is a simple curve. The rectiliear crossig umber cr (G) is the miimum umber of edge crossigs whe G is draw i the plae usig straight segmets as edges. The crossig umbers have may applicatios to Discrete Geometry ad Computer Sciece, see for example [7] ad [9]. I this paper we study the problem of determiig cr (K ),wherek deotes the complete graph o vertices. For simplicity we write cr () =cr (K ). A equivalet formulatio of the problem is to fid the miimum umber of covex quadrilaterals determied by poits i geeral positio (o three poits o a lie). We metio here that cr (K ) = 3 4 was cojectured by Zarakiewicz [] ad Guy [3], ad there are (o-rectiliear) drawigs of K achievig this umber. Of course cr (K ) cr (K ) but from the exact values of cr () for [], it is kow that cr (K 8 ) < cr (K 8 ). Jese [6] ad Siger [0] were the first to settle cr () =Θ. I fact, sice cr (5) = the by a averagig argumet it is easy to deduce that cr () 5. This same idea was used by Brodsky et al [] whe they obtaied cr (0) = 6, to deduce cr () Later Aicholzer et al [] calculated cr () = 53 adusedthistogetcr () Very recetly Wager [], followig differet methods proved cr () O the other had Brodsky et al [] costructed rectiliear drawigs of K showig cr () I this paper we prove the followig theorem which gives as a lower boud for cr () the exact value cojectured by Zarakiewicz ad Guy for cr (K ). Theorem cr() 4 3. It is kow that c =lim cr () / > 0 exists. Our theorem gives c 3/8 =0.375 ad it ca i fact be geeralized to a larger class of drawigs of K. Namely, those obtaied from the cocept of simple allowable sequeces of permutatios itroduced by Goodma ad Pollack [4]. We deote by P the real projective plae, a pseudolie ` is a simple closed curve whose removal does ot discoect P.Afiite set P i the plae is a geeralized cofiguratio if it cosists of a set of poits, together with a set of pseudolies joiig each pair of poits subject to the coditio that each pseudolie itersects every other exactly oce. If there is a sigle pseudolie for every pair the the geeralized cofiguratio is called simple.

2 Cosider a drawig of K i the (projective) plae where each edge is represeted by a simple curve. If each of these edges ca be exteded to a pseudolie i such a way that the resultig structure is a simple geeralized cofiguratio the we call such a drawig a pseudoliear drawig of K.Wecallpseudosegmets the edges of a pseudoliear drawig. Clearly, every rectiliear drawig of K is also pseudoliear. Thus the umber ecr(), defied as the miimum umber of edge crossigs over all pseudoliear drawigs of K, geeralizes the quatity cr() ad satisfies ecr() cr(). I this cotext we prove the followig stroger result. Theorem ecr() 3 4. If a pseudoliear drawig is combiatorially equivalet to a rectiliear drawig the it is called stretchable. It is kow that almost all pseudoliear drawigs are o-stretchable. So it is coceivable that ecr() < cr() for sufficietly large, but at the momet we have o other evidece to support this. We also metio that the problem of determiig whether a pseudoliear drawig is stretchable is NP-hard [8]. Allowable Sequeces Give a set P of poits i the plae, o three of them colliear, we costruct the + matrix S (P ) as follows. Cosider ay circle C cotaiig P i its iterior. Let ` be the vertical right-had side taget lie to C. We ca assume without loss of geerality that o segmet i P is perpedicular to `, we ca also assume that o two segmets i P are parallel, otherwise we ca perturb the set P without chagig the structure of its crossigs. Label the poits of P from to accordigtotheorderof their projectios to `, beig the lowest ad the highest. For each segmet ij i P,letc ij = c ji be the poit i the upper half of C such that the taget lie to C at c ij is perpedicular to ij. This gives a liear order o the segmets of P, iherited from the couter-clockwise order of the poits c ij i C. Deotebyt r the r th pair of poits (segmet) i P uder this order. Idistictly we use t r to deote a uordered pair {i, j} or the poit c ij = c ji. Usig this, we recursively costruct the matrix S (P ). The first row is (,,..., ), adthe(k +) th row is obtaied from the k th row by switchig the pair t k. S(P ) is half a period of what is commoly referred as a circular sequece of permutatios of P [4]. S (P ) satisfies the followig properties.. The first row of S (P ) is the -tuple (,, 3,..., ), the last row of S (P ) is the -tuple (,,...,, ), ad ay row of S (P ) is a permutatio of its first row.. Ay row r is obtaied from the previous row by switchig two cosecutive etries of the row r. 3. If the r th row is obtaied by switchig the etries S r,c ad S r,c+ i the (r ) th row the S r,c <S r,c+. 4. For every i<j thereexistsauiquerow r such that the etries i ad j are switched from row r to row r +, i.e., t r = {i, j}, S r,c = i<j<s r,c+,ad S r+,c = j>i= S r+,c+ for some c. A simple allowable sequece of permutatios is a combiatorial abstractio of a circular sequece of permutatios associated with a cofiguratio of poits. It is defied as a doubly ifiite periodic sequece of permutatios of,,..., satisfyig that every permutatio is obtaied from the previous oe by switchig two adjacet umbers, ad after i ad j have bee switched they do ot switch agai util all other pairs have switched. For the purposes of this paper we oly use half a period of a allowable sequece. This traslates to ay + matrix S (P ) satisfyig properties

3 -4. From ow o S(P ) will be such a matrix, ot ecessarily obtaied as the circular sequece of permutatios of a poit set P. It was proved by Goodma ad Pollack [5] that every simple allowable sequece of permutatios ca be realized by a geeralized cofiguratio of poits where the matrix S(P ) is determied by the cyclic order i which the coectig pseudolies meet a distiguished pseudolie (for example the pseudolie at ifiity). Next we establish whe two pseudosegmets do ot itersect by meas of the matrix S(P ). Give a simple geeralized cofiguratio of poits P, we say that two pseudosegmets e ab ad e cd are separated if there exists a pseudolie i P that leaves e ab ad e cd i differet sides. Note that ay two o-icidet pseudosegmets (i.e., they do ot share edpoits), either itersect i their iterior (geerate a crossig) or are separated. Thus ecr(g P )= ecr(p ) is the umber of o-icidet pairs of pseudosegmets mius the umber of separated pseudosegmets, where G P is a pseudoliear drawig of K associated to S(P ). Let < r be the liear order o {,, 3,...,} iduced by the r th row of S(P ). Observe that e ab ad e cd are separated if ad oly if there is a row r such that a, b < r c, d or c, d < r a, b. Ithiscase we say e ab ad e cd are separated i row r. Lemma 3 allows us to cout the umber of separated pseudosegmets i P. We say e ab ad e cd are eighbors i row r if they are separated i row r but ot i row r. Lemma 3 e ab ad e cd are separated if ad oly if there is a uique row r where e ab ad e cd are eighbors. Proof. Firstotethatifab e ad cd e are eighbors, the they are separated by defiitio. Now assume ab e ad cd e are separated, ad let R bethelastrowwheretheyareseparated. Ifab e ad cd e are separated i all rows above R the they are separated i the first ad cosequetly i the last rows, that is R = +. This is impossible sice havig ab e ad cd e separated i every row implies that they ever reversed their order. Cosider the largest row r R such that ab e ad cd e are ot separated i row r. The ab e ad cd e are eighbors i row r. Fially, to prove that such a row is uique, let r 0 <r be two rows where ab e ad cd e are eighbors. Assume without loss of geerality that a< r0 b< r0 c< r0 d. The a< r0 c< r0 b< r0 d ad, sice b ad c switch exactly oce, b< r c.also,bydefiitio, oe of the pairs eac, ad, f or bd e switches from row r to row r. Sice such a pair switches exactly oce, the it has opposite orders i rows r 0 ad r. Therefore oe of the followig should be satisfied b< r c< r a< r d, or b< r d< r a< r c, or a< r d< r b< r c, but the e ab ad e cd are ot separated i row r. ³ For all i 6= j i P,writef P ij e =(r, c), ifi ad j switch i row r ad colum c, thatis S r,c = i = S r+,c+ ad S r,c+ = j = S r+,c. Notethatthisiswelldefied sice the relative order of each pair of poits {i, j} i P is chaged exactly oce. For c defie C P (c) = r : there exist i, j such that f P ij e =(r, c), ad let ch P (c) =ch (c) = C P (c). I other words deotes the umber of chages (switches) i colum c. Lemma 4 For ay simple geeralized cofiguratio P of poits i the plae µ ecr (P )=3 (j ) ( j) ch (j). 4 3

4 Proof. Sice each four poits i P determie three pairs of o-icidet pseudosegmets, there are 3 4 pairs of o-icidet pseudosegmets i P. It remais to prove that (j ) ( j) ch (j) of these pairs are separated (o-crossig). Note that ab e ad cd e are eighbors i row r if ad oly if there are x {a, b}, y {c, d} such that x ad y switch from row r to row r. By Lemma 3, if t r = {i, ³ j} ad i<jthe all pairs hj f ad ik e are eighbors (i row r) wheever h< r j ad i< r k. If f P ij e =(r, c) the row r accouts for (c ) ( c) eighborig pairs of pseudosegmets. Moreover, Lemma 3 guaratees that, whe addig these quatities over all rows, we are coutig all separated pairs of pseudosegmets exactly oce. 3 Proof of Theorem Note that for fixed i, i switches exactly oce with each umber j 6= i, thatis ³ f P ij e : j, j 6= io =. Moreover, sice is the last etry i row ad the first etry i row +,thewhei = these switches occur i differet colums, that is ½ ³ c :f P j f =(r, c) for some r µ ¾,ad j< = {,,..., }. Therefore we ca defie R P (c) =r to be³ the uique row r where the chage of i colum c occurs, i.e., there exists j<such that f P j f =(r, c). Alsofor c defie the umber of chages i colum c above ad below row R P (c) as A P (c) = r<r P (c) :there exist i, j such that f P ij e =(r, c) B P (c) = r>r P (c) :there exist i, j such that f P ij e =(r, c). The proof of the Theorem is based o the idetity from Lemma 4, together with the ext two lemmas. Let m = b/c Lemma 5 For ay simple geeralized cofiguratio P of poits i the plae ad k m we have A P (k) + B P ( k) k. Proof. For j k let g (j) = mi r : there exists i such that f P ij e =(r, k) ³ o h (j) = mi r : there exists i such that f P ij e =(r, k). Sice all g (),g(),..., g (k),h(),h(),..., h (k) are differet, ad A P (k) ad B P ( k) are disjoit, the it is eough to prove that for all j k, eitherh (j) B P ( k) or g (j) A P (k). Assume that h (j) / B P ( k). The, sice h (j) 6= R P ( k), h (j) <R P ( k). Observe that g (j) <h(j) ad R P ( k) <R P (k) the Therefore g (j) A P (k). g (j) <h(j) <R P ( k) <R P (k). 4

5 Lemma 6 For ay simple geeralized cofiguratio P of poits i the plae ad k m we have µ k + (ch P (c)+ch P ( c)) 3( k) =3. Proof. By iductio o P =. The statemet is true for P =3by vacuity. Cosider the matrix S (P ) ad let P 0 = P {}. NotethatS (P 0 ) is the matrix obtaied from erasig the uique etry equal to i each row of S (P ) ad shiftig oe colum left the ecessary elemets of S (P ). Also the rows where the correspodig chage ivolves are deleted. Note that for c C P 0 (c) =A P (c) B P (c +). Thus for c Also otice that ad for c The by defiitio ad (3) ch P 0 (c) = A P (c) + B P (c +). () B P () = A P ( ) =. () ch P (c) = A P (c) + B P (c) +. (3) (ch P (c)+ch P ( c)) = ( A P (c) + B P (c) + A P ( c) + B P ( c) +) = k + ( A P (c) + B P (c) + A P ( c) + B P ( c) ), separatig oe term from each sum we get k (ch P (c)+ch P ( c)) = k + A P (k) + B P () + ( A P (c) + B P (c +) )+ the by () ad (), + A P ( ) + B P ( k) + k (ch P (c)+ch P ( c)) = k + A P (k) + B P ( k) + ch P 0 (c)+ ( A P ( c) + B P ( c +) ), c= ch P 0 ( c) k = k + A P (k) + B P ( k) + (ch P 0 (c)+ch P 0 ( c)). Fially, by iductio ad Lemma 5, (ch P (c)+ch P ( c)) k + k +3( (k )) c= µ k + = 3( k) =3. 5

6 ProofofTheorem.By Lemma 4, it is eough to fid a upper boud for the expressio (c ) ( c) ch P (c). For j m let x j = ch P (j) +ch P ( j), adx m = ch P (m) +ch P (m +) if is odd, otherwise x m = ch P (m). Uderthesedefiitios ad accordig to Lemma 5, together with the fact that P m x j =,itiseoughtofid the maximum of the fuctio f(x,x,...,x m )= m (j ) ( j) x j subject to the followig liear coditios: m µ µ k + x j = ad x j 3 for every k m. It is easy to see that the maximum occurs if ad oly if x k x m = 3 m.ifthisisthecasethe =3k for all k m ad ( 64 ( 3) ( ) 7 3 if is odd f(x,x,...,x m )= 64 ( ) if is eve. Therefore, by Lemma 5, we coclude that ecr (P ) 64 ( 3) ( ) if is odd 64 ( ) ( 4) if is eve. i.e., ecr (P ) j k ¹ º¹ º¹ º 3. 4 Refereces [] O. Aicholzer, F. Aurehammer, ad H. Krasser. O the crossig umber of complete graphs. I Proc.8thAACMSympCompGeom., Barceloa Spai, 9-4, 00. [] A. Brodsky, S. Durocher, ad E. Gether. Toward the Rectiliear Crossig Number of K :New Drawigs, Upper Bouds, ad Asymptotics. Discrete Mathematics. [3] R. K. Guy. The declie ad fall of Zarakiewicz s theorem, i Proc. Proof Techiques i Graph Theory, (F. Harary ed.), Academic Press, N.Y., 63-69, 969. [4] J. E. Goodma ad R. Pollack. Semispaces of cofiguratios, cell complexes of arragemets i P. J. Combi. Theory Ser. A, 3: -9, 98. [5] J. E. Goodma ad R. Pollack. A combiatorial versio of the isotopy cojecture. I J. E. Goodma, E. Lutwak, J. Malkevitch, ad R. Pollack, editors, Discrete Geometry ad Covexity, pages -9, volume 440 of A. New York Acad. Sci.,

7 [6] H. F. Jese. A upper boud for the rectiliear crossig umber of the complete graph. J. Combi. Theory Ser B, 0: -6, 97. [7] J. Matoušek. Lectures o Discrete Geometry. Spriger-Verlag, New York, N.Y., 00. [8] N.E. Mëv. O maifolds of combiatorial types of projective cofiguratios ad covex polyhedra. Soviet Math. Dokl., 3: , 985. [9] J. Pach ad G. Tóth. Thirtee problems o crossig umbers. Geombiatorics, 9: 94-07, 000. [0] D. Siger. Rectiliear crossig umbers. Mauscript, 97. [] U. Wager. O the Rectiliear Crossig Number of Complete Graphs. [] K. Zarakiewicz. O a problem of P. Turá cocerig graphs, Fud. Math. 4: 37-45,