Let α be a permutation of the vertex set V(G) of a connected graph G. Define the total relative displacement of α in G by δα(G) = ∑x,y∈ V(G) |dG(x,y)-dG(α(x), α(y))|, where dG(x, y) is the length of the shortest path between x and y in G. Let π*(G) be the maximum value of δα(G) among all permutations of V(G). The permutation which realizes π*(G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem is reduced to a quadratic integer programming problem. We characterize its optimal solution and present an algorithm running in O(n5 log n) time, where n is the total number of vertices in a complete multipartite graph.
- Chaotic mapping
- Complete multipartite graph
- Optimal solution
- Quadratic integer programming