Geodesic
On 3-simplicial vertices in planar graphs
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 3, pp. 413-421

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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.
Keywords: planar graph, outerplanar graph, 3-simplicial vertex 
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Boros, Endre; Jamison, Robert; Laskar, Renu; Mulder, Henry. On 3-simplicial vertices in planar graphs. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 3, pp. 413-421. http://geodesic.mathdoc.fr/item/DMGT_2004_24_3_a4/