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Discussiones Mathematicae Graph Theory 29(2) (2009) 385-400
DOI: https://doi.org/10.7151/dmgt.1454

ON LOCAL STRUCTURE OF 1-PLANAR GRAPHS OF MINIMUM DEGREE 5 AND GIRTH 4

Dávid Hudák  and  Tomás Madaras 

Institute of Mathematics, Faculty of Sciences
University of P. J. Safárik
Jesenná 5, 040 01 Košice, Slovak Republic
e-mail: davidh@centrum.sk, tomas.madaras@upjs.sk

Abstract

A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains

(1) a 5-vertex adjacent to an ≤ 6-vertex,

(2) a 4-cycle whose every vertex has degree at most 9,

(3) a K1,4 with all vertices having degree at most 11.

Keywords: light graph, 1-planar graph, star, cycle.

2000 Mathematics Subject Classification: 05C10.

References

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Received 13 November 2007
Revised 28 July 2008
Accepted 28 July 2008