×

zbMATH — the first resource for mathematics

Neighborhoods of edges in normal maps. (Russian) Zbl 0856.05031
A finite map represented on an oriented surface is said to be normal if every vertex has degree at least three and every edge is included in a face with a least three edges. The type of an edge \(e = xy\) in such a representation is the 4-tuple consisting of the degrees of \(x\) and \(y\) and the number of edges in the faces containing \(e\), written in a nondecreasing order. By definition the type \((t_1 t_2 t_3 t_4)\) is bounded above by the type \((t_1' t_2' t_3' t_4')\) if \(t_i \leq t_i'\) for every \(1 \leq i \leq 4\). In this paper it is proved that for every normal map on the torus there exists an edge whose type is bounded above by \((3 3 3 \infty)\), \((3 3 4 10)\), \((3 3 5 7)\), \((3 3 6 6)\), \((3 4 4 6)\) or \((4 4 4 4)\) and the property is strong; the property holds also for every normal map on an orientable surface of genus \(g\) having more than \(576 (g - 1)\) edges for 4-tuples: \((3 3 3 \infty)\), \((3 3 4 11)\), \((3 3 5 7)\), \((3 3 6 6)\), \((3 4 4 6)\) and \((4 4 4 4)\).

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
PDF BibTeX XML Cite