# On 3-simplicial vertices in planar graphs

Endre Boros; Robert E. Jamison; Renu Laskar; Henry Martyn Mulder

Discussiones Mathematicae Graph Theory (2004)

- Volume: 24, Issue: 3, page 413-421
- ISSN: 2083-5892

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topEndre Boros, et al. "On 3-simplicial vertices in planar graphs." Discussiones Mathematicae Graph Theory 24.3 (2004): 413-421. <http://eudml.org/doc/270616>.

@article{EndreBoros2004,

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.},

author = {Endre Boros, Robert E. Jamison, Renu Laskar, Henry Martyn Mulder},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {planar graph; outerplanar graph; 3-simplicial vertex},

language = {eng},

number = {3},

pages = {413-421},

title = {On 3-simplicial vertices in planar graphs},

url = {http://eudml.org/doc/270616},

volume = {24},

year = {2004},

}

TY - JOUR

AU - Endre Boros

AU - Robert E. Jamison

AU - Renu Laskar

AU - Henry Martyn Mulder

TI - On 3-simplicial vertices in planar graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2004

VL - 24

IS - 3

SP - 413

EP - 421

AB - 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.

LA - eng

KW - planar graph; outerplanar graph; 3-simplicial vertex

UR - http://eudml.org/doc/270616

ER -

## References

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- [2] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). Zbl0541.05054
- [3] R.E. Jamison and H.M. Mulder, Tolerance intersection graphs on binary trees with constant tolerance 3, Discrete Math. 215 (2000) 115-131, doi: 10.1016/S0012-365X(99)00231-9. Zbl0947.05055
- [4] B. Grünbaum and T.S. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Canad. J. Math. 15 (1963) 744-751, doi: 10.4153/CJM-1963-071-3. Zbl0121.37605
- [5] H. Lebesgue, Quelques conséquences simples de la formule d'Euler, J. Math. Pures Appl. 19 (1940) 27-43. Zbl0024.28701

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