Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T14:05:45.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Short proof of Menger's graph theorem

Published online by Cambridge University Press:  26 February 2010

G. A. Dirac
G. A. Dirac
Affiliation:
Trinity College, Dublin 2.
Get access

Extract

The object of this paper is to give a simple proof of Menger's famous theorem [1] for undirected and for directed graphs. Proofs of this theorem have been given by D. König [2], G. Nöbeling [3], G. Hajós [4], T. Gallai [5], P. Erdös [6] and 0. Ore [7]. The present proof is shorter, and formulated to apply to directed and undirected graphs equally. The term, graph is to be understood to mean either a finite undirected graph or a finite directed graph throughout. (A: B) denotes either an undirected edge between the two vertices A and B or a directed edge from A to B according to whether undirected or directed graphs are considered. G1 and G2 being two non-empty disjoint graphs, G1: G2-edge denotes an undirected edge between a vertex of G1 and a vertex of G2 or a directed edge from a vertex of G1 to a vertex of G2 as the case may be, and G1: G2-path denotes a path with one end-vertex in G1 and one in G2 having no intermediate vertex in G1 + G2, undirected or directed from the end in G1 to the end in G2, as the case may be. (A set of isolated vertices is a graph.)

Type
Research Article
Information
Mathematika , Volume 13 , Issue 1 , June 1966 , pp. 42 - 44
Copyright
Copyright © University College London 1966

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Menger, K., “Zur allgemeinen Kurventheorie”, Fund. Math., 10 (1927), 96.Google Scholar
2. König, D., “Über trennende Knotenpunkte in Graphen”, Acta Lit. Sci. Szeged, 6 (1933), 155.Google Scholar
3. Menger, K., Kurventheorie (Leipzig 1932), 221–228.Google Scholar
4. Hajós, G., “Zum Mengerschen Graphensatz”, Acta Lit. Szeged, 7 (1934), 44.Google Scholar
5. Grünwald, T., “Ein neuer Beweis eines Mengersehen Satzeal”, Journal London Math. Soc., 13 (1938), 188.CrossRefGoogle Scholar
6. König, D., Theorie der endlichen und unendlichen Graphen (Leipzig 1936), 247–248.Google Scholar
7. Ore, O.. Theory of graphs (Providence R. I., 1962), 200–202.Google Scholar
8. Dirae, G. A., “Extensions of Menger's theorem, Journal London Math. Soc., 38 (1963) 148.Google Scholar