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On removable cycles through every edge. (English) Zbl 1009.05083

Summary: W. Mader [Abh. Math. Semin. Univ. Hamb. 42, 187-204 (1974; Zbl 0266.05111)] and B. Jackson [J. Lond. Math. Soc., II. Ser. 21, 385-392 (1980; Zbl 0438.05038)] independently proved that every 2-connected simple graph \(G\) with minimum degree at least four has a removable cycle, that is, a cycle \(C\) such that \(G\setminus E(C)\) is 2-connected. This paper considers the problem of determining when every edge of a 2-connected graph \(G\), simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of two edges from \(G\) remains 2-connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge-disjoint removable cycles such that each contains one of the distinguished edges.

MSC:

05C38 Paths and cycles
05B35 Combinatorial aspects of matroids and geometric lattices
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[1] and Graph Theory with Applications, North-Holland, New York, 1976. · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2
[2] Fleischner, J London Math Soc 31 pp 193– (1985)
[3] Goddyn, J Combin Theory Ser B 71 pp 130– (1997)
[4] Goddyn, Combin Probab Comput 8 pp 539– (1999)
[5] Jackson, J London Math Soc 21 pp 385– (1980)
[6] Lemos, J Graph Theory 30 pp 51– (1999)
[7] Mader, Abh Math Sem Univ Hamburg 42 pp 187– (1974)
[8] Matroid Theory, Oxford University Press, New York, 1992.
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