Geodesic
The Existence Of P≥3-Factor Covered Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1055-1065

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A spanning subgraph F of a graph G is called a P≥3-factor of G if every component of F is a path of order at least 3. A graph G is called a P≥3-factor covered graph if G has a P≥3-factor including e for any e ∈ E(G). In this paper, we obtain three sufficient conditions for graphs to be P≥3-factor covered graphs. Furthermore, it is shown that the results are sharp.
Keywords: P≥3-factor, P≥3-factor covered graph, toughness, isolated toughness, regular graph 
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Zhou, Sizhong; Wu, Jiancheng; Zhang, Tao. The Existence Of P≥3-Factor Covered Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1055-1065. http://geodesic.mathdoc.fr/item/DMGT_2017_37_4_a14/