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 degF (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) ≤ Σx∈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.
For a set S of connected graphs, a spanning subgraph F of a graph is called an S-factor if every component of F is isomorphic to a member of S. It was recently shown that every 2-connected cubic graph has a {Cₙ | n ≥ 4}-factor and a {Pₙ | n ≥ 6}-factor, where Cₙ and Pₙ denote the cycle and the path of order n, respectively (Kawarabayashi et al., J. Graph Theory, Vol. 39 (2002) 188-193). In this paper, we show that every connected cubic bipartite graph has a {Cₙ | n ≥ 6}-factor, and has a {Pₙ | n ≥ 8}-factor if its order is at least 8.
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