Abstract
Publisher Summary A tournament is an orientation of a complete graph. Any two vertices (players) v, w are adjacent by exactly one arc, either v → w (v beats w) or w → v. Every tournament Tn has a Hamiltonian path—that is, every tournament Tn realizes the sequences 1, . . . , 1 and 0,. . . , 0. Various other binary sequences are known to be always realizable. Certain families of tournaments, most notably tournaments with n = 2k players, realize all possible 2n–1 binary sequences Bn–1. There are n distinct permutations of the players of Tn and only 2n–1 distinct patterns; these facts led the author to conjecture that every tournament Tn, n > 7, realizes every binary sequence Bn–1. This chapter discusses the evolution of this conjecture, the known results and generalizations to oriented trees, general digraphs, orientations of n-chromatic graphs, and complexity of computation.
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