By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Inst. Hungar. Acad. Sci. 7 (1962) 225-226. Annals of Discrete Mathematics 3 (1978) 49-53. @ North-Holland Publishing Company THE CHROMATIC INDEX OF THE GRAPH OF THE ASSIGNMENT POLYTOPE* Richard A. A. 1. Introduction Let n be a positive integer, and let S, denote the set of permutations of (1,. . , n}. We define a graph G, as follows. The set of vertices of G,, is S,. TWO vertices u,T E S, are joined by an edge in G, if and only if the permutation U-'T has exactly one non-trivial cycle (that is, a cycle of length at least two).

2. Theorem. If K: can be decomposed into hamiltonian cycles then K:, also can be decomposed into hamiltonian cycles. and Thus we have obtained a decomposition of the edges of K : , not of the form y’). 2 to decompose these remaining triples. Indeed with the directed hamiltonian cycle x l , . . -C. Bermond decomposition of K z we associate the following hamiltonian cycle of K i : . . (1,- 1 x:-,x,)(x,x:x (x,x ;x,)(x,x;x,) I) (x: x 1 x;) ~ . (x ;-, x, -1x A)( x ;x,x 1). Thus the proof is complete.

K ) . Instead of G" we consider a graph G' of n - 1 V,,l vertices, obtained from G" - V,, by omitting all the edges joining vertices from the same V, ( i = 1, . . , k ) ; (ii) joining a V, to a V, for an "exceptional pair", that is, (*) does not hold; (iii) joining a V, to a V,, when d ( V , , V,)

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

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