Geodesic
Star-Cycle Factors of Graphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 1, pp. 193-198

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A spanning subgraph F of a graph G is called a star-cycle factor of G if each component of F is a star or cycle. Let G be a graph and f : V (G) → 1, 2, 3, . . . be a function. Let W = v ∈ V (G) : f(v) = 1. Under this notation, it was proved by Berge and Las Vergnas that G has a star-cycle factor F with the property that (i) if a component D of F is a star with center v, then deg_F (v) ≤ f(v), and (ii) if a component D of F is a cycle, then V (D) ⊆ W if and only if iso(G − S) ≤ Σ_x∈S f(x) for all S ⊂ V (G), where iso(G − S) denotes the number of isolated vertices of G − S. They proved this result by using circulation theory of flows and fractional factors of graphs. In this paper, we give an elementary and short proof of this theorem.
Keywords: star factor, cycle factor, star-cycle factor, factor of graph 
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Egawa, Yoshimi; Kano, Mikio; Yan, Zheng. Star-Cycle Factors of Graphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 1, pp. 193-198. http://geodesic.mathdoc.fr/item/DMGT_2014_34_1_a15/