Geodesic
Linear forests and ordered cycles
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 3, pp. 359-372

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A collection L = P¹ ∪ P² ∪ ... ∪ 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.
Keywords: hamilton cycles, graph linkages 
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     title = {Linear forests and ordered cycles},
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Chen, Guantao; Faudree, Ralph; Gould, Ronald; Jacobson, Michael; Lesniak, Linda; Pfender, Florian. Linear forests and ordered cycles. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 3, pp. 359-372. http://geodesic.mathdoc.fr/item/DMGT_2004_24_3_a0/