A pair of forbidden subgraphs and 2-factors

In this paper, we consider pairs of forbidden subgraphs that imply the existence of a 2-factor in a graph. For d ≥ 2, let G _d be the set of connected graphs of minimum degree at least d. Let F ₁ and F ₂ be connected graphs and let H be a set of connected graphs. Then {F ₁, F ₂} is said to be a forbidden pair for H if every {F ₁, F ₂}-free graph in H of sufficiently large order has a 2-factor. Faudree, Faudree and Ryjáček have characterized all the forbidden pairs for the set of 2-connected graphs. We first characterize the forbidden pairs for G ₂, which is a larger set than the set of 2-connected graphs, and observe a sharp difference between the characterized pairs and those obtained by Faudree, Faudree and Ryjáček. We then consider the forbidden pairs for connected graphs of large minimum degree. We prove that if {F ₁, F ₂} is a forbidden pair for G _d, then either F ₁ or F ₂ is a star of order at most d + 2. Ota and Tokuda have proved that every K _1,, ⌊d+2/2⌋-free graph of minimum degree at least d has a 2-factor. These results imply that for k ≥ d + 2, no connected graphs F except for stars of order at most d + 2 make {K _1,k, F} a forbidden pair for G _d, while for k≤ ⌊d+2/2⌋ every connected graph F makes {K _1,k, F} a forbidden pair for G _d. We consider the remaining range of ⌊ d + 2/2⌋ < k < d+2, and prove that only a finite number of connected graphs F make {K _1,k, F} a forbidden pair for G _d.