Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups |
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Authors: | Štefko MiklaviČ Primož Šparl |
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Institution: | 1. Primorska Institute for Natural Science and Technology, University of Primorska, , Koper, Muzejski Trg 2, 6000, Slovenia;2. Faculty of education, University of Ljubljana, , Kardeljeva Plo??ad 16, 1000 Ljubljana, Slovenia |
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Abstract: | In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ is n‐HC‐extendable if it contains a path of length n and if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ is weakly n‐HP‐extendable if it contains a path of length n and if every such path is contained in some Hamilton path of Γ. Moreover, Γ is strongly n‐HP‐extendable if it contains a path of length n and if for every such path P there is a Hamilton path of Γ starting with P. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory |
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Keywords: | Hamilton cycle Hamilton path n‐HC‐extendable strongly n‐HP‐extendable weakly n‐HP‐extendable Cayley graph abelian group |
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