DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 25(3) (2005) 419-426
DOI: 10.7151/dmgt.1293

ON SHORT CYCLES THROUGH PRESCRIBED VERTICES OF A POLYHEDRAL GRAPH

Erhard Hexel

Department of Mathematics
Technische Universität Ilmenau
Postfach 0565, D-98684 Ilmenau, Germany

Abstract

Guaranteed upper bounds on the length of a shortest cycle through k ≤ 5 prescribed vertices of a polyhedral graph or plane triangulation are proved.

Keywords: polyhedral graph, triangulation, short cycle, prescribed vertices.

2000 Mathematics Subject Classification: 05C38.

References

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[2] G.A. Dirac, 4-crome Graphen und vollständige 4-Graphen, Math. Nachr. 22 (1960) 51-60, doi: 10.1002/mana.19600220106.
[3] F. Göring, J. Harant, E. Hexel and Zs. Tuza, On short cycles through prescribed vertices of a graph, Discrete Math. 286 (2004) 67-74, doi: 10.1016/j.disc.2003.11.047.
[4] J. Harant, On paths and cycles through specified vertices, Discrete Math. 286 (2004) 95-98, doi: 10.1016/j.disc.2003.11.059.
[5] R. Diestel, Graph Theory (Springer, Graduate Texts in Mathematics 173, 2000).
[6] A.K. Kelmans and M.V. Lomonosov, When m vertices in a k-connected graph cannot be walked round along a simple cycle, Discrete Math. 38 (1982) 317-322, doi: 10.1016/0012-365X(82)90299-0.
[7] T. Sakai, Long paths and cycles through specified vertices in k-connected graphs, Ars Combin. 58 (2001) 33-65.

Received 3 September 2004
Revised 18 February 2005