# Advances in Graph Theory by B. Bollobás (Eds.)

By B. Bollobás (Eds.)

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**Example text**

N } we choose the element defined by the choice design of order n. For the triples we choose (i, j , a ) (i,j , P ) (i, j , Y) with i < j , i y if i + j = O (mod3), j P i if i + j = l (mod3), a i j if i + j = 2 (mod3). For the triples we choose (i, a, S) (i, a, Y) (2, P, Y) 1 Y y P a 1 a i P if i E 0 (mod 3), if i = 1 (mod 3), if i = 2 (mod3). For the triple ( a , p, y ) we choose y. We leave to the reader the care of checking that we obtain a choice design of order n + 3; the only non-immediate part is to check property (ii) for the triples containing a pair (i, a ) or (i, p ) or (i, y ) .

The cycle in the thick line is longer than L 46 B. 7, A. Hobbs Let W = V(G)-A. By Lemma 3 we have IAl= m - k and so I W(= m + k. Put GI = G[ W]. Order the vertices in W as w,, w 2 , . . ,so that deg, (w,)>deg, (wi) whenever i

6. 3. If n is odd, there exists a decomposition of K , into i ( n- 1) hamiltonian cycles. To each of these cycles (x,, x2, . . ,x,) we associate the following $ ( n-2) hamiltonian cycles of K:: (xl, y;, x2)(x2, y;, x3) . . 5). The set {y; :i = 1 , 2 , . . ,;(n -2)) consists of the ;(n-2) elements representatives of the $(n -2) triples (x,, x,,,~, y) containing the pair {x,, x,,,} and where neither x, nor x , , ~ has been chosen. Thus we have constructed i ( n - l)(n -2) hamiltonian cycles of K : and it suffices to verify that no triple (edge) appears twice, but that follows from the definition of a choice design of order n.