Boros, Endre; Jamison, Robert E.; Laskar, Renu; Mulder, Henry Martyn On 3-simplicial vertices in planar graphs. (English) Zbl 1068.05018 Discuss. Math., Graph Theory 24, No. 3, 413-421 (2004). Authors’ abstract: A vertex \(v\) in a graph \(G=(V,E)\) is \(k\)-simplicial if the neighborhood \(N(v)\) of \(v\) can be vertex-covered by \(k\) or fewer complete graphs. The main result of the paper states that a planar graph of order at least four has at least four 3-simplicial vertices of degree at most five. This result is a strengthening of the classical corollary of Euler’s formula that a planar graph of order at least four contains at least four vertices of degree at most five. Reviewer: Bela Andrásfai (Budapest) MSC: 05C10 Planar graphs; geometric and topological aspects of graph theory 05C75 Structural characterization of families of graphs PDFBibTeX XMLCite \textit{E. Boros} et al., Discuss. Math., Graph Theory 24, No. 3, 413--421 (2004; Zbl 1068.05018) Full Text: DOI