×

zbMATH — the first resource for mathematics

A Kotzig type theorem for non-orientable surfaces. (English) Zbl 1141.05028
The weight of an edge in a graph \(G\) is the sum of the degrees of its end vertices. The weight of \(G\), \(w(G)\), is the minimum weight among all its edges. The authors show that for a graph with minimum degree at least 3 which is embeddable in a non-orientable surface of genus \(q \geq 1\) the following holds: \(w(G)\leq 2q+11\) for \(1\leq q\leq 2\), \(w(G)\leq 2q+9\) for \(3\leq q\leq 5\), otherwise \(w(G)\leq 2q+7\); and the bounds are best possible.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] IVANČO J.: The weight of a graph. Ann. Discrete Math. 51 (1992), 113-116. · Zbl 0773.05066
[2] JENDROL’ S.-VOSS H.-J.: Light subgraphs of graphs embedded in \(2\)-dimensional manifolds of Euler characteristic \(< 0\) - a survey. P. Erdos and his Mathematics II. (G. Halasz, L. Lovasz, M. Simonovits, V. T. Sos, Bolyai Soc. Math. Stud. 11, Springer, Budapest, 2002, pp. 375-411. · Zbl 1037.05015
[3] JENDROL’ S.-VOSS H.-J.: A local property of large polyhedral maps on compact \(2\)-dimensional manifolds. Graphs Combin. 15 (1999), 303-313. · Zbl 0933.05044
[4] KOTZIG A.: A contribution to the theory of Eulerian polyhedra. Mat.-Fyz. Časopis SAV (Math. Slovaca) 5 (1955), 101-113.
[5] MOHAR B.-THOMASSEN C.: Graphs on Surfaces. The Johns Hopkins University Press, Baltimore-London, 2001. · Zbl 1230.05133
[6] RINGEL G.: Der vollständige paare Graph auf nichtorientierbaren Flächen. J. Reine Angew. Math. 220 (1965), 89-93. · Zbl 0132.21204
[7] RINGEL G.: Map Color Theorem. Springer-Verlag, Berlin, 1974. · Zbl 0287.05102
[8] THOMASSEN C.: Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface. Trans. Amer. Math. Soc. 323 (1991), 605-635. · Zbl 0722.05031
[9] ZAKS J.: Extending Kotzig’s Theorem. Israel J. Math. 45 (1983), 281-296. · Zbl 0524.05031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.