Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

Abstract

AbstractIn this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ isn‐HC‐extendableif it contains a path of lengthnand if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ isweakly n‐HP‐extendableif it contains a path of lengthnand if every such path is contained in some Hamilton path of Γ. Moreover, Γ isstrongly n‐HP‐extendableif it contains a path of lengthnand if for every such pathPthere is a Hamilton path of Γ starting withP. 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