## DIMACS TR: 2001-32

## On 3-Simplicial Vertices in Planar Graphs

### Authors: Endre Boros, Robert E. Jamison, Renu Laskar and Henry Martyn Mulder

**
ABSTRACT
**

A vertex $v$ in a graph $G = (V,E)$ is {\em $k$-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.

Paper Available at:
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