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More about the weight of edges in planar graphs. (English) Zbl 0854.05039
Define the weight of an edge of graph $$G$$ to be the sum of the degrees of its vertices. Let $$w$$ denote the minimum weight of edges in $$G$$. Let $$w^*$$ denote the minimum weight of edges that are incident with at least one triangle. Let $$w^{**}$$ denote the minimum weight of edges that are incident with at least two triangles. Clearly, $$w\leq w^*\leq w^{**}$$. The results of the paper apply to planar maps $$G$$ in which every face and every vertex is incident with at least three edges. The paper shows that if $$w^{**}= \infty$$, then either $$w^*\leq 9$$ or $$w\leq 8$$, and both bounds are sharp.
Reviewer: M.Marx (Pensacola)

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C35 Extremal problems in graph theory
##### Keywords:
planar graphs; weight; planar maps; face; bounds