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Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V (G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore's theorem which guarantees the existence of a Hamilton path connecting any… (More)

Let G be a graph and f : V (G) → {1, 3, 5,. . .}. Then a spanning subgraph F of G is called a (1, f)-odd factor if deg F (x) ∈ {1, 3,. .. , f(x)} for all x ∈ V (G). We give some results on (1, f)-odd factors and k-critical graphs with respect to (1, f)-odd factor. We consider finite graphs without loops or multiple edges. Let G be a graph with vertex set V… (More)