# 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
##### Keywords:
oriented surface; type of an edge; normal map; torus; genus