By Claude Berge, etc.

ISBN-10: 0444865128

ISBN-13: 9780444865120

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Put C = {u,,,. . , and P = {ue, u I , . . , u,}. As 1 is odd, if uoE X I , ur E X z and I r - ( u , J nPI = IP n X , I = h - s , Ir+(u,)nPI = I P n X I (= k - r . Since 1 + c = 2 ( h + k ) - 3 , we can suppose that the numbering of C is such that for any i E [0, I ] , u , E f - ( u O ) ,and u , E T ' ( s ) : r+(ur) n c, ~= ~r -+ (~~n ), ,c. , . , U , + ~ , -=~ J { u ~ -u~~, -. ~. ,,u ~ - P) Fig. 1. I odd. Fig. 2. I even. J. Aye1 36 As f-~(uo) n P = P fl X 2 , uI E f - ( u o ) and (u", uI, . .

For every n and p . we give the minimum number of e d g e s in the digraph D of order rt, insuring the existence of D ( n , p ) in D. In this paper, we consider digraphs with no loop or multiple edges. We use standard terminology (see Berge [I]). However, we specify below some definitions and notations. , the pair of edges (x, y ) and ( y , x ) , ( x , y ) is one of edges ( x , y ) or ( y , x ) (it can be symmetric); the edge ( x , y ) is called antisymmetric if ( y , x ) E E ( D ) . 4 is the subdigraph of D induccd by V ( D ) - A .

A,,in which . ,anoccur on the diagonal outside A if A is of type I or type 11. 0 degree n-r 0 c1 degree n-r a’ a’ r+l f deeree n-r-2 0 0 0 0 degree n-r-I ‘J’ n-1 ‘J’ 0 0 Fig. 7. G , ( A ) and G , ( A ) for a type I1 square A (with k 0 =n and I =r + I). C r Small embeddings 27 Proof. Let A be as stated and let uk,ul be as in the definition of type I and type 11. Suppose that A is embeddable. Then A has an extension to an embeddable square A ' of side r + 1 in which uk occurs in cell ( r + 1, r + 1).