2-factors in 1-tough k-connected graphs with prescribed independence number

In 1972, Chvátal and Erdös proved the classic result that any graph whose independence number is at most its vertex-connectivity is Hamiltonian. Let G be a graph with independence number α(G) and vertex-connectivity κ(G). If α(G)>κ(G), then the Chvátal-Erdös proximity of G, denoted by CE(G), is defined as CE(G)=α(G)−κ(G). In this paper we discuss the existence of 2-factors in 1-tough graphs with CE(G)=1,2, or 3. In particular, we prove that a 1-tough graph G with CE(G)=ℓ has a 2-factor, unless it belongs to a specified family of graphs. These are improvements within this class over the best possible result that every 2-tough graph on at least 3 vertices has a 2-factor. Finally, we present a topic for further research.