Geodesic
The structure and existence of 2-factors in iterated line graphs
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 507-526

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We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where L^k(G) contains a 2-factor.
Keywords: line graph, 2-factor, iterated line graph, cycle 
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     title = {The structure and existence of 2-factors in iterated line graphs},
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Ferrara, Michael; Gould, Ronald; Hartke, Stephen. The structure and existence of 2-factors in iterated line graphs. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 507-526. http://geodesic.mathdoc.fr/item/DMGT_2007_27_3_a7/