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 Gd be the set of connected graphs of minimum degree at least d. Let F1 and F2 be connected graphs and let H be a set of connected graphs. Then {F1, F2} is said to be a forbidden pair for H if every {F1, F2}-free graph in H of sufficiently large order has a 2-factor. Faudree, Faudree and Ryjáček [Discrete Math. 308 (2008), 1571–1582] have characterized all the forbidden pairs for the set of 2-connected graphs. We first characterize the forbidden pairs for G2, 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 et al. We then consider the forbidden pairs for connected graphs of large minimum degree. We prove that if {F1, F2} is a forbidden pair for Gd, then either F1 or F2 is a star of order at most d + 2. Ota and Tokuda [J. Graph Theory 22 (1996), 59–64] have proved that every K1,⌊ d+2 2 ⌋-free graph of minimum degree at ∗Partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), 22740068, 2010 †Partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 22500018, 2010