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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: madaras@science.upjs.sk

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.

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Received 28 January 2003
Revised 16 April 2004