Abstract

. Walecki tournaments were dened by Alspach in 1966. They form a class of regular tournaments that posses a natural Hamilton directed cycle decomposition. It has been conjectured by Kelly in 1964 that every regular tournament possesses such a decomposition. Therefore Walecki tournaments speak in favor of the conjecture. A second interest in Walecki tournaments arises from the mapping between cycles of the complementing circular shift register and isomorphism classes of Walecki tournaments. An upper bound on the number of isomorphism classes of Walecki tournaments was determined by Alspach. It was conjectured that the bound is tight. The problem of enumerating Walecki tournaments has not been solved to date. However, it was published as an open problem in a paper by Alspach in 1989. In an attempt to prove this 34 years old conjecture, we rst determine the arc structure of Walecki tournaments for all initial cases and those whose corresponding binary sequences have zero pattern. Subsequent papers deal with more general cases. Techniques used in

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