2- and 3-factors of Graphs on Surfaces

It has been conjectured that any 5-connected graph embedded in a surface Σ with sufficiently large face-width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4-connected graphs. In this paper, we shall study the existence of 2and 3-factors in a graph embedded in a surface Σ. A hamiltonian cycle is a special case of a 2-factor. Thus it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2-factor. In fact, we only need the 4-connectivity with minimum degree at least 5. In addition, our face-width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface Σ of Euler genus g. (1) If G is 4-connected and minimum degree of G is at least 5, and furthermore, face-width of G is at least 4g − 12, then G has a 2-factor. (2) If G is 5-connected and face-width of G is at least max{44g − 117,5}, then G has a 3-factor. The connectivity condition for both results are best possible. In addition, the face-width conditions are necessary too.