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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.

05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
Full Text: DOI
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