Cycles in multipartite tournaments: results and problems
Discrete Mathematics
Pancyclic orderings of in-tournaments
Discrete Applied Mathematics - Sixth Twente Workshop on Graphs and Combinatorial Optimization
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An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k-pancyclicity of in-tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in-tournament is vertex k-pancyclic for k ≤ 5 and k ≥ n - 3. In the latter case, we even show that the in-tournaments in consideration are fully (n - 3)-extendable which means that every vertex belongs to a cycle of length n - 3 and that the vertex set of every cycle of length at least n - 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in-tournament of order n with minimum degree greater than ${{9(n-k-1)}\over{5+6k+(-1)^k2^{-k+2}}}+1$ is vertex k-pancyclic for 5 k n - 3, and we present a family of examples showing that this bound would be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 84–104, 2001