[ a , B ]-Factors of Graphs on Surfaces

A well-known conjecture of Grünbaum (1970) and Nash-Williams (1971) asserts that every 4-connected toroidal graph has a Hamiltonian cycle. Related to this conjecture, Kawarabayashi and Ozeki (2011) proved two results on a 2-factor and a 3-factor. In this paper, motivated by these results, we give several sufficient conditions for a graph embedded in a surface to have an [ a , b ]-factor. We also show that several conditions are best possible.