Mod 3 Arithmetic on Triangulated Riemann Surfaces.

Edward G. Belaga (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let $T$ be a triangulation of a Riemann surface, orientable or non-orientable and of an arbitrary genus. Suppose, a labelling of the vertices of $T$ by three labels 0, +1, and -1 is fixed. The present paper deals with the following problem : find the number of labellings 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 labelling of faces is equivalent, according to P. J. Heawood, to the {it 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 $T$.

Mots clés: Riemann surface; Heawood vector; "Riemann surface; triangulationgraph; four-colouring; Heawood vector"

Domaines: Combinatoire [math.CO]

Edward G. Belaga. Mod 3 Arithmetic on Triangulated Riemann Surfaces.. 1999. ⟨hal-00129632⟩