By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Theorem. If K: can be decomposed into hamiltonian cycles then K:, also can be decomposed into hamiltonian cycles. (This proof was obtained with D. ) In order to shorten the writing, I will write a hamiltonian cycle as E,,. . E,, . . ,En where EL= (xE,y,, IC,+,). Let the vertex set of K:, be X U X’ with = [X’l=n. With each hamiltonian cycle of the decomposition of K: we associate 4 hamiltonian cycles of K : , in the following manner: 1x1 (i) if n is even, we associate with (x,y,x,)(x,y,x,), . .

Raspail, 75006 Paris, Cidex 06, France 0. Definitions Definitions not given here can be found in [3]. We will say that a graph G (undirected or directed) can be decomposed into Hamiltonian cycles or paths if we can partition its edges (arcs in the directed case) into hamiltonian cycles or paths (directed cycles or directed paths in the directed case). Our notation is as follows: K , -the complete graph on n vertices; K*,-the complete symmetric directed graph on n vertices; K,,, -the complete r-partite graph whose vertex set is the disjoint union of r sets of n elements, two vertices being joined iff they belong to two different sets; C, (resp.

Example. Let ( 0 , 1 , 2 , 3 , 4 ) and ( 0 , 2 , 4 , 1 , 3 ) be two hamiltonian cycles of K,; by using the choice design of the example we obtain the two hamiltonian cycles of K:: If n is even the proof is similar. 2). To each of these directed cycles (xl, x2, . . ,: y;, xJ(x2, Y;, ~ 3 . ) . (xn-19 ~ i - 1 xn)(xn, , Y:, where i = 1 , 2 , . . , i ( n - 2 ) ; y j f y‘; for i f k and where the set { y i : i = 1 , 2 , . . ,b(n - 2) is determined as follows. Consider the $(n- 2) elements representative of the triples (x,, x,+,, y ) containing the pair {x,, x,,,} and where neither x, nor x , , ~ has been chosen.

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


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