By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Posa, A theorem concerning Hamiltonian lines, Publ. Math. 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).

Generally, if x p and W,+, are already defined, we check whether the set X , = {x,, . . , x p } represents all the L c A,. Bollobas, P. Simonovits, E. Szemeridi so that xPcl is not joined to W,, at all. At the end of the procedure we have an X = X , and a W'= W,,, not joined at all to each other. Let B ' c B , be the class of vertices of degree at least i n -+onT-' in A , . Let D be the set of vertices of A, joined to B' completely. D is relatively large. Indeed, by the minimum property of the partition ( A , ,A,) any x E B' is joined to at least n($- & T I ) vertices of A,, hence at least n($+ F ) - Tn(&iir' + E ) vertices of A, are joined to B' completely.

Let c' < c be fixed. We shall say that the edges are regularly distributed, if for every partition V ( G " ) = A U B we have d ( A , B ) 3 2 c ' . If we have an arbitrary G", we shall find a G"' in it, in which the edges are regularly distributed and h(G"') 3 cm2,m > n; also hold, where h ( G )denotes the minimum number of edges one has to omit to change G into a bipartite graph. Therefore it will be enough to prove the theorem for the case, when the edges are regularly distributed and this will be just Part 11.

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

by Thomas

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