Solution of a conjecture of Volkmann on longest paths through an arc in strongly connected in‐tournaments |
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Authors: | Dirk Meierling |
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Institution: | Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany |
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Abstract: | An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. Let m = 4 or m = 5 and let D be a strongly connected in‐tournament of order such that each arc belongs to a directed path of order at least m. In 2000, Volkmann showed that if D contains an arc e such that the longest directed path through e consists of exactly m vertices, then e is the only arc of D with that property. In this article we shall see that this proposition is true for , thereby validating a conjecture of Volkmann. Furthermore, we prove that if we ease the restrictions on the order of D to , the in‐tournament D in question has at most two such arcs. In doing so, we also give a characterization of the in‐tournaments with exactly two such arcs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 130–148, 2009 |
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Keywords: | In‐tournament longest path m‐path arc |
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