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On Independent Circuits Contained in a Graph

Published online by Cambridge University Press:  20 November 2018

p. Erdös  and
L. Pósa
p. Erdös
Affiliation:
University of British Columbia, Vancouver, and Budapest, Hungary
L. Pósa
Affiliation:
University of British Columbia, Vancouver, and Budapest, Hungary
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A family of circuits of a graph G is said to be independent if no two of the circuits have a common vertex; it is called edge-independent if no two of them have an edge in common. A set of vertices will be called a representing set for the circuits (for the sake of brevity we shall call it a representing set), if every circuit of G passes through at least one vertex of the representing set. Denote by I(G) = k the maximum number of circuits in an independent family and by R(G) the minimum number of vertices of a representing set. Dirac and Gallai asked whether there is any relation between I(G) and R(G) (trivially R(G) ≥ I(G)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 347 - 352
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Erdös, P. and Posa, L., On the maximal number of disjoint circuits of a graph, Publ. Math. Debrecen, 9 (1962), 3–12.Google Scholar
2. Erdös, P., Graph theory and probability, Can. J. Math., 11 (1959), 34–8.Google Scholar
3. Erdös, P., On circuits and subgraphs of chromatic graphs, Mathematika, 9 (1962), 170–5.Google Scholar
4. Gallai, T., Maximum-minimum Sàtze and verallgemeinerte Faktoren von Graphen. Acta Math. Hung. Acad. Sci., 12 (1961), 131–73; cf. pp. 158 and 161.Google Scholar