A theorem on tait colorings with an application to the generalized Petersen graphs

Abstract

A family of graphs which includes the Petersen graph is postulated, and it is conjectured that the Petersen graph is the only member of this family not to have a Tait coloring. A general theorem about Tait colorings is proved and the conjecture is shown to be equivalent to a combinatorial assertion involving cyclically ordered arrays of n objects each belonging to one of 3 distinguishable classes. Finally, the combinatorial formulation is used to show that the conjecture is true wherever the parameters of the family satisfy any of a number of equalities or congruences.