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On light edges and triangles in planar graphs of minimum degree five. (English) Zbl 0813.05020
Let $$G$$ be a planar graph of minimum degree five. Denote by $$e_{i,j}$$ the number of edges of $$G$$ joining a vertex of degree $$i$$ to a vertex of degree $$j$$, and by $$f_{i,j,k}$$ the number of faces of $$G$$ which have exactly one vertex of degree $$i$$, $$j$$, and $$k$$, respectively, in their boundary. The first result states that $$7e_{5,5}+ 3e_{5,6}\geq 180$$, with the coefficients being best possible. Secondly, an example is provided to show that the coefficients in a previous inequality of Borodin, namely $18 f_{5,5,5}+ 9 f_{5,5,6}+ 5 f_{5,5,7}+ 4f _{5,6,6}\geq 144,$ are best possible.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C15 Coloring of graphs and hypergraphs
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##### References:
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