DMGT

Discussiones Mathematicae Graph Theory 22(1) (2002) 193-198
DOI: https://doi.org/10.7151/dmgt.1168

TRESTLES IN POLYHEDRAL GRAPHS

Michal Tkáč

Department of Mathematics
The Faculty of Business Economics in Košice
University of Economics in Bratislava
Tajovskeho 13, 041 30 Košice, Slovakia
e-mail: mtkac@euke.sk

Heinz-Jürgen Voss

Institute of Algebra
Technical University Dresden
Mommsenstrasse 13, D-01062 Dresden, Germany
e-mail: voss@math.tu-dresden.de

Abstract

A k-trestle of a graph G is a 2-connected spanning subgraph of G of maximum degree at most k. We show that a polyhedral graph G has a 3-trestle, if the separator-hypergraph of G contains no two different cycles joined by a path of 3-separators of length ≥ 0. There are graphs not satisfying this condition that have no 3-trestles. Further, for each integer k every graph with toughness smaller than [2/k] has no k-trestle.

Keywords: polyhedral graphs, non-Hamiltonian, k-trestle.

2000 Mathematics Subject Classification: Primary 05C38, Secondary 52B10.

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Received 24 July 2000
Revised 19 July 2001