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

[BIBTex] [PDF] [PS]

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

[1] O.V. Borodin, Solution of Kotzig-Grünbaum problems on separation of a cycle in planar graphs (Russian), Mat. Zametki 46 (1989) 9-12; translation in Math. Notes 46 (1989) 835-837, doi: 10.1007/BF01139613.
[2] O.V. Borodin, Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs, Metody Diskret. Analiz. 41 (1984) 12-26 (in Russian).
[3] O.V. Borodin, A new proof of the 6 color theorem, J. Graph Theory 19 (1995) 507-521, doi: 10.1002/jgt.3190190406.
[4] O.V. Borodin, A.V. Kostochka, A. Raspaud and E. Sopena, Acyclic coloring of 1-planar graphs (Russian), Diskretn. Anal. Issled. Oper. 6 (1999) 20-35; translation in Discrete Appl. Math. 114 (2001) 29-41.
[5] Z.-Z. Chen and M. Kouno, A linear-time algorithm for 7-coloring 1-plane graphs, Algorithmica 43 (2005) 147-177, doi: 10.1007/s00453-004-1134-x.
[6] R. Diestel, Graph Theory (Springer, New York, 1997).
[7] I. Fabrici and T. Madaras, The structure of 1-planar graphs, Discrete Math. 307 (2007) 854-865, doi: 10.1016/j.disc.2005.11.056.
[8] S. Jendrol' and T. Madaras, On light subgraphs in plane graphs of minimum degree five, Discuss. Math. Graph Theory 16 (1996) 207-217, doi: 10.7151/dmgt.1035.
[9] S. Jendrol', T. Madaras, R. Soták and Z. Tuza, On light cycles in plane triangulations, Discrete Math. 197/198 (1999) 453-467, doi: 10.1016/S0012-365X(99)90099-7.
[10] V.P. Korzhik, Minimal non-1-planar graphs, Discrete Math. 308 (2008) 1319-1327, doi: 10.1016/j.disc.2007.04.009.
[11] G. Ringel, Ein Sechsfarbenproblem auf der Kügel, Abh. Math. Sem. Univ. Hamburg 29 (1965) 107-117, doi: 10.1007/BF02996313.
[12] H. Schumacher, Zur Structur 1-planar Graphen, Math. Nachr. 125 (1986) 291-300.

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