The Heawood approach to Tait colorings and defining vertex sets

Abstract

Given a simple biconnected planar cubic graph, we associate each its vertex among $2n$ ones with the so-called spin, i.e., a variable which takes on values $\pm 1$. P. J. Heawood has proved that a Tait coloring, accurate to the choice of a color for one edge, is equivalent to the choice of spin values so as to make the sum of these value at vertices of any face be a multiple of~3. We treat faces, which satisfy this condition, as {\it proper}. The condition that guarantee the propriety of faces define a system of linear equations (SLE) with respect to variables, which take on nonzero values in the field ${\mathbb F}_3$. We say that a set of vertices is {\it defining} if values of spins of these vertices uniquely define values of the rest spins. In particular, so is the set of vertices which correspond to all free variables of the SLE. We actualize the approach proposed by P. J. Heawood by proposing a geometric proof of the fact that for a non-bipartite graph the rank of the SLE equals $n+1$. Moreover, we also geometrically describe the necessary condition for the minimality of the defining set. This implies that in the case of a non-bipartite graph there exist defining subsets consisting of $n-1$ vertices. As a simple corollary, we conclude that the number of Tait colorings in this case does not exceed $3\cdot 2^{n-1}$. Though this estimate is not exact, it is by half better than the known one. We also prove that the number of Tait colorings for a graph $CL_n$, which is bipartite for even $n$ and non-bipartite for an odd one, equals $2^n+8$ and $2^n-2$, correspondingly.