Sanders, Daniel P. On circuits through five edges. (English) Zbl 0864.05054 Discrete Math. 159, No. 1-3, 199-215 (1996). 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proved the conjecture true for \(k=3\) while the case \(k=4\) was settled affirmatively, independently, by Robertson (unpublished manuscript), P. L. Erdös and E. Györi [Acta Math.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 \(k=5\). Reviewer: R.C.Entringer (Albuquerque) Cited in 2 Documents MSC: 05C38 Paths and cycles Keywords:independent set; circuit; conjecture Citations:Zbl 0362.05069; Zbl 0588.05024; Zbl 0751.05059 PDFBibTeX XMLCite \textit{D. P. Sanders}, Discrete Math. 159, No. 1--3, 199--215 (1996; Zbl 0864.05054) Full Text: DOI 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 Hungar., 4, 82 (1974) [7] N. Robertson, unpublished manuscript.; N. Robertson, unpublished manuscript.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 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.