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On circuits through five edges. (English) Zbl 0864.05054

L. Lovász [Problem 5, Period. Math. Hung. 4, 82 (1974)] and D. R. Woodall [J. Comb. Theory, Ser. B 22, 274-278 (1977; Zbl 0362.05069)] independently conjectured that if \(L\) is an independent set of \(k\) vertices in a \(k\)-connected graph \(G\), then \(G\) has a circuit containing the edges of \(L\) iff \(L\) is not an odd edge cut. Lovász [op. cit.] proved the conjecture true for \(k=3\) while the case \(k=4\) was settled affirmatively, independently, by Robertson (unpublished manuscript), P. L. Erdös and E. Györi [Acta Math. Hung. 46, 311-313 (1985; Zbl 0588.05024)], and M. V. Lomonosov [Cycles through prescribed elements in a graph, Paths, flows, and VLSI-layout, Proc. Meet., Bonn/Ger. 1988, Algorithms Comb. 9, 215-234 (1990; Zbl 0751.05059)]. The author shows the conjecture is true for the case \(k=5\).

MSC:

05C38 Paths and cycles
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References:

[1] Aldred, R. E.L.; Holton, D. A.; Thomassen, C., Cycles through four edges in 3-connected cubic graphs, Graphs and Combin., 1, 7-11 (1985) · Zbl 0717.05043
[2] Dirac, G. A., In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen, Math. Nachr., 22, 61-85 (1960) · Zbl 0096.17903
[3] Erdös, P. L.; Györi, E., Any four independent edges of a 4-connected graph are contained in a circuit, Acta Math. Hungar., 46, 311-313 (1985) · Zbl 0588.05024
[4] Häggkvist, R.; Thomassen, C., Circuits through specified edges, Discrete Math., 41, 29-34 (1982) · Zbl 0488.05048
[5] Lomonosov, M. V., Cycles through prescribed elements in a graph, (Korte; Lovász; Prömel; Schrijver, Paths, Flows, and VLSI Layout (1990), Springer: Springer Berlin), 215-234 · Zbl 0751.05059
[6] Lovász, L., Problem 5, Period. Math. Hungar., 4, 82 (1974)
[7] N. Robertson, unpublished manuscript.; N. Robertson, unpublished manuscript.
[8] Thomassen, C., Note on circuits containing specified edges, J. Combin. Theory Ser. B, 22, 279-280 (1977) · Zbl 0364.05031
[9] Woodall, D. R., Circuits containing specified edges, J. Combin. Theory Ser. B, 22, 274-278 (1977) · Zbl 0362.05069
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