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JOURN AL OF COMBINATORIAL THEORY 9, 324--326 (1970)

O n No n r e c o n s t r u c t a b le T o u r n a m e n t s *L. W. BEINEKE AND E. T. PARKER

Purdue University at Fort Wayne, Indiana 46805,and University of lllinois, Urbana, Illinois 61801Communicated by Frank Harary

Rece ived De cem ber 7 , 1968

ABSTRACTP a i r s o f n o n - i s o m o rp h ic st ro n g to u rn a m e n t s o f o rd e r s 5 a n d 6 a re give n fo rw h ic h th e s u b to u rn a m e n t s o f o rd e r s 4 a n d 5 , r es p e ct ive ly , a re p a i rw is e i s o mo r -p h ic . H e re to fo re , o n ly p a i rs o f o rd e r s 3 a n d 4 w e re k n o w n .

A tournament o f o r d e r n h a s n n o d e s w i t h e x a c t l y o n e d i r e c t e d a rcj o i n i n g e a c h p a i r o f d i s t in c t n o d e s ; i n o t h e r w o r d s , i t i s a c o m p l e t e ,i r r ef l e x iv e , a s y m m e t r i c r e l a t i o n o n a s e t o f n e l e m e n t s . A t o u r n a m e n t T o fo r d e r n c l e a r ly d e t e r m i n e s n s u b t o u r n a m e n t s o f o r d e r n - - 1, e a c h o b t a i n e df r o m T b y d e l e t i n g o n e n o d e a n d t h e i n c i d e n t a rc s . A p r o b l e m i n t h e s t u d yo f t o u r n a m e n t s , s o m e t i m e s c a l l e d U l a m ' s p r o b l e m i n a n a l o g y w i t h o n e i nu n d i r e c t e d g r a p h s , h a s b e e n c o n s i d e r e d b y H a r a r y a n d P a l m e r [1 ] a n dc a n b e s t a te d a s f o l lo w s : F o r w h a t p a i r s o f n o n - i s o m o r p h i c t o u r n a m e n t so f o r d e r n a r e th e t w o s et s o f n s u b t o u r n a m e n t s o f o r d e r n - - 1 p a i r w i s e -i s o m o r p h i c ? A l t e r n a t i v e l y , g i v en n t o u r n a m e n t s o f o r d e r n - - 1 , w h e n c a nn o n - i s o m o r p h i c t o u r n a m e n t s o f o r d e r n b e r e c o n s t r u c t e d f r o m t h e m ?

A t o u r n a m e n t i s c a l l e d strong i f t h e r e i s n o p a r t i t i o n o f it s n o d e s i n t on o n - e m p t y s u b se t s U a n d V s u c h t h a t a l l a r c s g o f r o m U t o V .A w e l l - k n o w n t h e o r e m s ta t es t h a t a t o u r n a m e n t i s s t r o n g i f a n d o n l y i f i th a s a s p a n n i n g d i r e c t e d c y cl e . H a r a r y a n d P a l m e r s h o w e d t h a t , i f t w on o n - i s o m o r p h i c t o u r n a m e n t s o f o r d e r n > 4 a r e n o t s t r o n g , th e n t h e i rs u b t o u r n a m e n t s o f o r d e r n - - 1 c a n n o t b e p a i r w i s e - is o m o r p h i c . I t s h o u l db e n o t e d ( s ee M o o n [ 2, p . 3 ] ) t h a t " a l m o s t a l l " t o u r n a m e n t s a r e s t r o n g ,h o w e v e r .

* Resea rch o f f irs t au tho r sup por te d by A ir F orc e Off ice o f Sc ien ti fic Resea rchg ra n t A F O S R -6 8 -1 5 1 5 . R e s e a rc h o f s e c o n d a u th o r s u p p o r te d b y O f f i c e o f N a v a lResearch grant N000 14-67-A-0305-0008.3 2 4

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326 BE1NEKE AND PARKERTo see that the tour nam ent s of order 5 are non-isomorphic , we note

that, alt hough the list of scores (numb ers of out-goi ng arcs from the nodes)is (3, 2, 2, 2, 1) in both, in one the arc f ro m the n ode of score 1 goes to theone of score 3, while in the other the corresponding arc goes to a node ofscore 2. Since each of the two tournaments of order 6 has a uniquetransitive sub tou rna men t of order 5, they are clearly non-isomorphic.

There are no known larger examples; perhaps there are none. Or' isthere an example for every order ? Such conjectures and questions abound.By considering the converse to urna ment s (obtaine d by reversing thedirect ion of every arc), we observe that the pairs of tou rn am en ts of orders 4and 6 are converses of each other, while the tour na me nts of orders 3 an d 5are self-converse. Is this a p atte rn which might prove helpful ? In con-clusion we menti on the analogous pr oble m for directed graphs in general:Do examples exist which are not tou rna men ts ?

Added in proof Exhaustive examination of all 56 tournaments with six nodeshas yielded two additional pairs whose subtournaments of order 5 are pairwise-iso-morphic. These are shown in Figs. 5 and 6. In Fig. 5, the arc from the node with score 1goes to a node with score 2 in the first case, and the corresponding arc goes to a nodewith score 3 in the second. In Fig. 6, the three nodes having score 2 form a transitivetriple in one case, and a cyclic triple in the other. Thus, the two tournaments in eachfigure are non-isomorphic. That the proper subtournaments are pairwise-isomorphicin each case is a matter of routine verification. We note that the tournaments in eachpair are converses of each other.

FIG. 5. Order 6.

FIG. 6. Order 6.

REFERENCES1. F. HARARYAND E. PALMER,On the problem of reconstructing a tournament fromsubtournaments, Monatsh. Math. 71 (1967), 14-23.2. J. W. MOON,Topics on Tournaments, Holt, Rinehart, & Winston, New York, 1968.