Geodesic
The existence of path-factor covered graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 5-16

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A spanning subgraph H of a graph G is called a P_≥ k-factor of G if every component of H is isomorphic to a path of order at least k, where k≥2. A graph G is called a P_≥ k-factor covered graph if there is a P_≥ k-factor of G covering e for any e∈ E(G). In this paper, we obtain two special classes of P_≥ 2-factor covered graphs. We also obtain two special classes of P_≥ 3-factor covered graphs. Furthermore, it is shown that these results are all sharp.
Keywords: path-factor, $P_{\geq2}$-factor covered graph, $P_{\geq3}$-factor covered graph, claw-free graph, isolated toughness 
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Dai, Guowei. The existence of path-factor covered graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 5-16. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a0/