Graph Theory, 1736-1936
Handbook of combinatorics (vol. 1)
Handbook of combinatorics (vol. 1)
Colouring, stable sets and perfect graphs
Handbook of combinatorics (vol. 1)
A first course in geometric topology and differential geometry
A first course in geometric topology and differential geometry
| Hi-index | 5.23 |
Let T be a triangulation of a Riemann surface, orientable or non-orientable and of an arbitrary genus. Suppose, a labeling of the vertices of T by three labels &Thgr;, +1, and -1 is fixed. The present paper deals with the following problem: 2nd the number of labelings of the faces of T by two labels +1 and - 1, in such a way that the sum of the labels of the faces around any vertex is equal modulo 3 to the given label of the vertex. If T is a planar triangulation and all labels of vertices are zeros, then the problem of existence of such a labeling of faces is equivalent, according to P.J. Heawood, to the four-colour problem for planar triangulations, and the corresponding counting problem is equivalent to that of counting the number of all proper four-colourings of T.