Mod 3 Arithmetic on Triangulated Riemann Surfaces.

Belaga, Edward G.

Belaga, Edward G.

HAL, hal-00129632 / Harvested from HAL

Text on HAL

Résumé

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

$T$ .

Publié le : 1999-05-20
Classification: Riemann surface, Heawood vector, "Riemann surface, triangulationgraph, four-colouring, Heawood vector", 05C, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]

@article{hal-00129632,
     author = {Belaga, Edward G.},
     title = {Mod 3 Arithmetic on Triangulated Riemann Surfaces.},
     journal = {HAL},
     volume = {1999},
     number = {0},
     year = {1999},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00129632}
}

Belaga, Edward G. Mod 3 Arithmetic on Triangulated Riemann Surfaces.. HAL, Tome 1999 (1999) no. 0, . http://gdmltest.u-ga.fr/item/hal-00129632/