Forbidden subgraphs and the existence of a 2-factor

For a connected graph H, a graph G is said to be H-free if G does not contain an induced subgraph which is isomorphic to H, and for a set of connected graphs H, G is said to be H-free if G is H-free for every H ∈ H. The set H is called a forbidden subgraph. In particular, if |H| = 2 (resp. |H| = 3), we often call H a forbidden pair (resp. forbidden triple). In 1991, Bedrossian proved that there essentially exist only three forbidden pairs that guarantee the existence of a hamiltonian cycle in 2-connected graphs. Then in 1997, Faudree and Gould proved that even if we relax the conclusion by allowing a finite number of exceptions, there essentially exist only four forbidden pairs. After these studies, forbidden triples for hamiltonicity have been investigated. But complete characterization has not yet been obtained. In this talk, we investigate forbidden subgraphs which guarantee the existence of a 2-factor in a sufficiently large connected graphs of minimum degree at least two. We will see that the problem is much simpler than that of a hamiltonian cycle. As a result, we obtain a complete characterization of forbidden pairs and triples.