On 3-simplicial vertices in planar graphs

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 3simplicial 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.