PDF

Discussiones Mathematicae Graph Theory 24(3) (2004) 403-411
DOI: https://doi.org/10.7151/dmgt.1239

ON THE STRUCTURE OF PLANE GRAPHS OF MINIMUM FACE SIZE

Tomás Madaras

Institute of Mathematics, Faculty of Sciences
University of P.J. Safárik
Jesenná 5, 041 54 Košice, Slovak Republic
e-mail: [email protected]

Abstract

A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is known that a plane graph of minimum face size 5 contains light paths and a light pentagon. In this paper we show that every plane graph of minimum face size 5 contains also a light star K_1,3 and we present a structural result concerning the existence of a pair of adjacent faces with degree-bounded vertices.

Keywords: plane graph, light graph, face size.

2000 Mathematics Subject Classification: 05C10.

References

[1]	H. Enomoto and K. Ota, Connected Subgraphs with Small Degree Sums in 3-Connected Planar Graphs, J. Graph Theory 30 (1999) 191-203, doi: 10.1002/(SICI)1097-0118(199903)30:3<191::AID-JGT4>3.0.CO;2-X.
[2]	I. Fabrici, On vertex-degree restricted subgraphs in polyhedral graphs, Discrete Math. 256 (2002) 105-114, doi: 10.1016/S0012-365X(01)00368-5.
[3]	I. Fabrici, J. Harant and S. Jendrol', Paths of low weight in planar graphs, submitted.
[4]	I. Fabrici, E. Hexel, S. Jendrol' and H. Walther, On vertex-degree restricted paths in polyhedral graphs, Discrete Math. 212 (2000) 61-73, doi: 10.1016/S0012-365X(99)00209-5.
[5]	I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs and Combin. 13 (1997) 245-250.
[6]	I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar graphs, Discrete Math. 191 (1998) 83-90, doi: 10.1016/S0012-365X(98)00095-8.
[7]	J. Harant, S. Jendrol' and M. Tkáč, On 3-connected plane graphs without triangular faces, J. Combin. Theory (B) 77 (1999) 150-161, doi: 10.1006/jctb.1999.1918.
[8]	S. Jendrol', T. Madaras, R. Soták and Z. Tuza, On light cycles in plane triangulations, Discrete Math. 197/198 (1999) 453-467.
[9]	S. Jendrol' and P. Owens, On light graphs in 3-connected plane graphs without triangular or quadrangular faces, Graphs and Combin. 17 (2001) 659-680, doi: 10.1007/s003730170007.
[10]	A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. Cas. SAV (Math. Slovaca) 5 (1955) 111-113.
[11]	H. Lebesgue, Quelques consequences simples de la formule d'Euler, J. Math. Pures Appl. 19 (1940) 19-43.
[12]	P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413-426, doi: 10.1007/BF01444968.

Received 28 January 2003
Revised 16 April 2004