SUR UN THEOREMS DU TYPE KÖNIG POUR HYPERGRAPHES

P ' P + l a r e a l l d i f f e r e n t v e r t i c e s , are a l l d i f f e r e n t edges , P P ( 1 ) X I , X 2 , . . . .x (2) E I s E 2 s * . s E ( 3 ) X k s X k + l C Ek (k = 1 , 2 ,..., p ) . I f p > I , and i f x = x l , w e have a cyc le o f l e n g t h p. P+ 1 A t r a n s v e r s a l set of H i s a se t TCX, such t h a t T n E i # @ ( i = l , 2 , ..., m). A pack ing i s a fami ly o f p a i r w i s e d i s j o i n t edges. The purpose of t h i s paper i s t o g e n e r a l i z e KSnig's theorem f o r b i p a r t i t e g raphs , and alsosome p r o p e r t i e s of unimodular m a t r i c e s , as fo l lows : A hypergraph H (and a l l i t s p a r t i a l sub-hypergraphs H I ) have a minimum t r a n s v e r s a l set and a maximum packing w i t h t h e same c a r d i n a l i t i e s if and only i f every 0dd c y c l e ( x l , E l , , . . , x ~ ~ + ~ ) of H p o s s e s s e s an edge Ei which c o n t a i n s a t l e a s t t h t e e v e r t i c e s of t h e cycle. This r e s u l t was found i n connec t ion wi th a c o n j e c t u r e on p e r f e c t g raphs . 40 Annals New Y ork Academy of Sciences R E F E R E N C E S -.-.-.-.-.-.-.-.-.. . . . . . . . . ( I ) C. BERGE : Some Classes of P e r f e c t Graphs. Graph Theory and Theor e t i c a l Phys ic s (Harary Ed. ) , Chap. 5, Academic P r e s s . 1967. (2) C. BERGE : The Rank of a Family o f Sets and some Appl i ca t ions t o Graph Theory. Recent P r o g r e s s i n Combinatorics, ( T u t t e Ed . ) , Academic P r e s s 1969, 49-57. (3) C. BERGE : Thgor ie des Graphes e t des Hypergraphes, Fondements (4) D.K. RAY CHAUDHURI : An Algorithm for a m a x i m u m cover of an absMathgmatiques. Chap. 17 e t 18. (To appear ) . t ract complex. Canad. J. of Math., 15, 1963, 11-24.