Forbidden subgraphs and the existence of a 2-factor

In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F ₂ be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph Gis said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)-F ₂ is a finite class. In particular, we investigate whether H must contain a star (i.e. K_1,n for some positive integer n). We prove the following. (1) If |H|=1, then H={K_1,2}. (2) If |H|=2, then H contains K_1,2 or K_1,3. (3) If |H|=3, then H contains a star. But no restriction is imposed on the order of the star. (4) Not all of H with |H|=4 contain a star. For |H|≤2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582), and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)-G({K_1,t}) is finite for some t≥3.