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Linear forests and ordered cycles

Guantao Chen;  Ralph J. Faudree;  Ronald J. Gould;  Michael S. Jacobson;  Linda Lesniak;  Florian Pfender — 2004

Discussiones Mathematicae Graph Theory

A collection L = P ¹ ∪ P ² ∪ . . . ∪ P t (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian.

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