Jendroľ, Stanislav; Madaras, Tomáš Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs. (English) Zbl 1150.05321 Tatra Mt. Math. Publ. 30, 149-153 (2005). The local structure of planar graphs is investigated. Let the edge weight \(w(e)\) of an edge \(e=uv \in E(G)\) be defined as \(w(e) = {\operatorname {deg}}(u)+{\operatorname {deg}}(v)\). It is proved that each planar graph of minimum degree 3 and minimum edge weight at least 9 contains a vertex of degree \(d\), \(d \in \{3,4,5\}\), which is adjacent with at least \(d - 1\) vertices of degree at most 20. The planar complete bipartite graphs \(K_{1,r}\) and \(K_{2,r}\), \(r \geq 3\) show that the requirement on minimum degree is necessary. An example, the kleetope of an icosahedron, where each vertex of degree 3 is adjacent to vertices of degree 20, is presented to show that the bound 20 is the best possible. In the proof a nontrivial recharging method is applied. Reviewer: Peter Mihók (Košice) Cited in 1 ReviewCited in 13 Documents MSC: 05C10 Planar graphs; geometric and topological aspects of graph theory Keywords:planar graph; weight of edge; local structure PDFBibTeX XMLCite \textit{S. Jendroľ} and \textit{T. Madaras}, Tatra Mt. Math. Publ. 30, 149--153 (2005; Zbl 1150.05321)