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Let be a permutation of V (G) of a connected graph G. De ne the total relativ e displacement of in G by (G) = X x;y2V (G) jd G (x; y) ? d G ((x); (y))j where d G (x; y) is the length of the shortest path between x and y in G. Let (G) be the maximum value of (G) among all permutations of V (G) and the permutation which realizes (G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem will reduce to a quadratic integer programming. We characterize its optimal solution and present an algorithm running in O(n 5 log n) time where n is the total number of vertices in a complete multipartite graph. key word. Chaotic mapping, complete multipartite graph. AMS(MOS) subject classi cation. 05C05

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