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Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in-tournaments

✍ Scribed by Dirk Meierling

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
110 KB
Volume
63
Category
Article
ISSN
0364-9024

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✦ Synopsis

Abstract

A directed cycle C of a digraph D is extendable if there exists a directed cycle C′ in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non‐Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3≤k<n: a digraph D is k ‐extendable if every directed cycle of length t, where k≤t<n, is extendable. Moreover, D is called fully k ‐extendable if D is k ‐extendable and every vertex of D belongs to a directed cycle of length k. An in‐tournament is an oriented graph such that the in‐neighborhood of every vertex induces a tournament. This class of digraphs which generalizes the class of tournaments was introduced by Bang‐Jensen, Huang and Prisner in 1993. Tewes and Volkmann showed that every connected in‐tournament D of order n with minimum degree δ≥1 is ($n- \lfloor {{4\delta+1} \over {3}} \rfloor$) ‐extendable. Furthermore, if D is a strongly connected in‐tournament of order n with minimum degree δ=2 or $\delta {>}{{8n-17}\over {31}}$, then D is fully ($n-\lfloor {{4\delta+1} \over {3}}\rfloor$) ‐extendable. In this article we shall see that if$3 \leq \delta \leq {{8n-17} \over {31}}$, every vertex of D belongs to a directed cycle of length $n-\lfloor {{4\delta+1} \over {3}}\rfloor$, which means that D is fully ($n-\lfloor {{4\delta+1} \over {3}}\rfloor$) ‐extendable. This confirms a conjecture of Tewes and Volkmann. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 82–92, 2010

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