A New Solution to the Intersection Problem of Mendelsohn Triple Systems. Rachel Watson

Transcription

1 A New Solution to the Intersection Prolem of Mendelsohn Triple Systems y Rchel Wtson A thesis sumitted to the Grdute Fculty of Auurn University in prtil fulfillment of the requirements for the Degree of Mster of Science Auurn, Alm My, 0 Keywords: Mendelsohn triple system, intersection prolem, cyclic triple Copyright 0 y Rchel Wtson Approved y Chrles Lindner, Chir, Professor of Mthemtics Chris Rodger, Professor of Mthemtics Peter Johnson, Professor of Mthemtics Den Hoffmn, Professor of Mthemtics

2 Astrct This thesis gives new nd much simpler proof of the intersection prolem for Mendelsohn triple systems. THE INTERSECTION PROBLEM: For ech n 0 or (mod ), n, determine the set of ll k such tht there exists pir of MTS(n) hving exctly k cyclic triples in common. In wht follows, we will set I[n] = {0,,,.., x = n(n ) } \ {x, x, x, x } nd denote y J[n] = {k there exists two MTS(n) hving k cyclic triples in common}. In [] it ws shown tht J[] = {}, J[] = {0, } nd J[n] = I[n] for ll n (n 0 or (mod ), of course). The ojective of this thesis is new nd much simpler proof of the intersection prolem using results completely different from those used in the originl solution. ii

3 Tle of Contents Astrct ii Introduction nd Outline The Intersection of Idempotent Qusigroups The Bsic Constructions The n Construction The nα Construction: The n + Construction: The Solution for n The Solution for n Concluding Remrks Biliogrphy iii

4 Chpter Introduction nd Outline The complete directed grph D n is the grph with n vertices in which ech pir of distinct vertices re joined y two directed edges in opposite directions. D n = We will denote the directed edge from to y (, ). A cyclic triple is collection of three directed edges of the form {(, ), (, c), (c, )} where,, nd c, re distinct. c We will denote this cyclic triple y ny cyclic shift of (,, c). Finlly, Mendelsohn Triple System (nmed fter N.S. Mendelsohn[]) of order n (MTS(n)) is pir (S, T ), where T is collection of edge disjoint cyclic triples which prtition D n with vertex set S.

5 Exmple. Two MTS (). Let S = {,,,,,, } nd T nd T e the following two MTS()s. T T D = D =

6 It is immedite tht the two MTS() in this exmple hve exctly one cyclic triple in common, nmely (,, ). (,, ) = It is well-known tht the spectrum for Mendelsohn Triple Systems is precisely the set of ll n 0 or (mod ) EXCEPT for n = (there does not exist Mendelsohn Triple System of order ), nd if (S, T ) is MTS(n), T = n(n ). THE INTERSECTION PROBLEM: For ech n 0 or (mod ), n, determine the set of ll k such tht there exists pir of MTS(n) hving exctly k cyclic triples in common. In wht follows, we will set I[n] = {0,,,.., x = n(n ) }\{x, x, x, x } nd denote y J[n] = {k there exists two MTS(n) hving k cyclic triples in common}. In [] it ws shown tht J[] = {}, J[] = {0, } nd J[n] = I[n] for ll n (n 0 or (mod ), of course). The ojective of this thesis is new nd much more simple proof of the intersection prolem using results completely different from those used in the originl solution.

7 Chpter The Intersection of Idempotent Qusigroups A qusigroup (Q, ) is sid to e idempotent provided x x = x for ll x Q. Two idempotent qusigroups re sid to intersect in k products provided their tles gree in exctly k cells off of the min digonl. Exmple. (Two idempotent qusigroups of order intersecting in products). In [] H.L. Fu proved the following theorem. Theorem. (H.L. Fu []). If n, there exists pir of idempotent qusigroups of order n hving k products in common if nd only if k {0,,,..., x = n n} \ {x, x, x, x }. We will use this result to give much simpler solution to the intersection prolem eginning with order 8.

8 Chpter The Bsic Constructions We give three sic constructions in this chpter which will e used for ll of the intersection results which follow.. The n Construction Let (Q, ) e n idempotent qusigroup of order n, set S = Q {,, }, nd define collection of cyclic triples T s follows:. ((, ), (, ), (, )) nd ((, ), (, ), (, )) T for ll Q; nd (, ) (, ) (, ). For ech Q the six cyclic triples ((, i), (, i), (, i + )), ((, i), (, i), (, i + )) T. ( ) ( ) ( ) ( )

9 Then, (S, T ) is MTS(n).. The nα Construction: Let Q = {,,,..., n}, α = (,,,..., n), nd (Q, ) n idempotent qusigroup of order n. Set S = Q {,, } nd define collection of cyclic triples T α s follows:. ((, ), (, ), (α, )) nd ((, ), (α, ), (, )) T α for ll Q; nd (, ) (, ) (α, ). For ech Q, the six cyclic triples ((, ), (, ), (, )), ((, ), (, ), (, )), ((, ), (, ), (( )α, )), ((, ), (, ), (( )α, )), ((, ), (, ), (( )α, )), ((, ), (, ), (( )α, )) elong to T α. ( )α ( )α ( )α ( )α Then (S, T α) is MTS(n).. The n + Construction: Let (Q, ) e n idempotent qusigroup of order n, set S = { } (Q {,, }), nd define collection of cyclic triples T s follows:

10 . For ech x Q, plce copy of C = {(,, ), (,, ), (,, ), (,, )} on { } ({x} {,, }); C (x, ) (x, ) (x, ) nd plce these cyclic triples in T ; nd. for ech Q, plce the six cyclic triples ((, i), (, i), (, i+)), ((, i), (, i), (, i + )) T. ( ) ( ) ( ) ( ) Then (S, T ) is MTS(n + ). With these three constructions in hnd, long with the results in Chpter, we cn give very simple nd elegnt solution to the intersection prolem for Mendelsohn triple systems eginning with n = 8.

11 Chpter The Solution for n + 9 This is the esier of the two equivlence clsses; so good plce to egin. Let (Q, ), (Q, ), (Q, ), (Q, ), (Q, ), (Q, ) e ny six idempotent qusigroups of order n. Further, let M nd M e the two Mendelsohn triple systems of order defined elow: Then M M =. M = {(,, ), (,, ), (,, ), (,, )}, nd M = {(,, ), (,, ), (,, ), (,, )}. Set S = { } (Q {,, }) nd define two MTS(n + )s T nd T s follows: T : (i) For ech x Q plce copy of M or M on { } ({x} {,, }) nd plce these cyclic triples in T. M or M (x, ) (x, ) (x, ) 8

12 (ii) For ech x y Q plce the six cyclic triples ((x, i), (y, i), (x i y, i + )) nd ((y, i), (x, i), (y i x, i + )) in T. x y x y y x x y x y y x x y x y x y Then (S, T ) is MTS(n). T : (i) For ech x Q plce copy of M or M on { } ({x} {,, }) nd plce these cyclic triples in T. M or M (x, ) (x, ) (x, ) 9

13 (ii) For ech x y Q plce the six cyclic triples ((x, i), (y, i), (x i y, i + )) nd ((y, i), (x, i), (y i x, i + )) in T. x y y x x y y x x x y y y x x x y y Then (S, T ) is MTS(n). It is immedite tht the intersection numer for (S, T ) nd (S, T ) is T T = n j= m+ k + k + k, where m {0, } nd (Q, ) (Q, ) = k, (Q, ) (Q, ) = k, (Q, ) (Q, ) = k. A stright-forwrd clcultion shows tht ny k I[n + ] cn e written in the form n j= m + k + k + k, where k, k, k {0,,,..., x = n n} \ {x, x, x, x }. Since J[n + ] I[n + ] ( necessry condition), it follows tht I[n + ] J[n + ] so tht I[n + ] = J[n + ]. We hve the following result. Lemm. J[n + ] = I[n + ] for ll n

14 Chpter The Solution for n 8 There re two cses to consider here: () k n, nd (not too surprisingly) () k n. ) k n. We will use the n nd nα Constructions here. Set S = Q {,, } nd let (Q, ) nd (Q, ) e pir of idempotent qusigroups of order n. Since n, for ny k {,,,..., x = n n}\{x, x, x, x }, we cn tke (Q, ) (Q, ) = k. It is importnt to note tht ny k n {0,,,..., x = n n}\{x, x, x, x }. Now define two MTS(n)s T nd T s follows: T : Use the n Construction with (Q, ). T : Use the nα Construction with (Q, ) from the first to the second level; nd (Q, ) etween the second nd third, nd third nd first levels.

15 ( )α ( )α α ( )α ( )α Clerly the intersection numer for T T is (Q, ) (Q, ) = k. ) k n. In this cse we use the n Construction with pirs of qusigroups s in the n + solutions. It is immedite tht ny k n I[n] cn e written in the form n+k +k +k where k, k, k {0,,,..., x = n n}\{x, x, x, x }. Since J[n] I[n], it follows tht I[n] J[n] nd I[n] = J[n]. We hve the following result. Lemm. J[n] = I[n] for ll n 8.

16 Chpter Concluding Remrks Comining Lemms. nd. gives the following result. Theorem. J[n] = I[n] for ll n 0 or MTS() exists. (mod ) 8, except n = for which no The solution for the cses where n cn e found in the originl pper [] nd re hndled y n eclectic collection of d-hoc constructions. A quick glnce t [] will convince the reder tht the solution for n 8 given in this thesis is vstly superior to the originl solution in its simplicity.

17 Biliogrphy [] Hng-Lin Fu, On the construction of certin types of ltin squres with prescried intersections, Ph. D. thesis, Auurn University, (980). [] D.G. Hoffmn nd C.C. Lindner, Mendelsohn triple systems hving prescried numer of triples in common, Europ. J. Comintorics, (98), -. [] N.S. Mendelsohn, A Nturl Generliztion of Steiner Triple Systems, in Computers in Numer Theory, (A.O.L. Atkin nd B.J. Birch, eds), Acdemic Press, London, 9, -8.