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Spanning cycles in regular matroids without small cocircuits. (English) Zbl 1248.05030
Summary: A cycle of a matroid is a disjoint union of circuits. A cycle \(C\) of a matroid \(M\) is spanning if the rank of \(C\) equals the rank of \(M\). Settling an open problem of D. Bauer [Congr. Numerantium 49, 11–18 (1985; Zbl 0621.05021)], P. A. Catlin [J. Graph Theory 12, No. 1, 29–44 (1988; Zbl 0659.05073)] showed that if \(G\) is a 2-connected graph on \(n>16\) vertices, and if \(\delta(G)>\frac {n}{5}-1\), then \(G\) has a spanning cycle. Catlin also showed that the lower bound of the minimum degree in this result is best possible. In this paper, we prove that for a connected simple regular matroid \(M\), if for any cocircuit \(D\), \(|D|\geq \) max\(\{\frac {r(M)-4}{5},6\}\), then \(M\) has a spanning cycle.

05B35 Combinatorial aspects of matroids and geometric lattices
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[1] Bauer, D., A note on degree conditions for Hamiltonian cycles in line graphs, Congr. numer., 49, 11-18, (1985)
[2] Boesch, F.T.; Suffel, C.; Tindell, R., The spanning subgraphs of Eulerian graphs, J. graph theory, 1, 79-84, (1977) · Zbl 0363.05042
[3] Bondy, J.A.; Murty, U.S.R., Graph theory, (2008), Springer New York · Zbl 1134.05001
[4] Catlin, P.A., A reduction method to find spanning Eulerian subgraphs, J. graph theory, 12, 29-44, (1988) · Zbl 0659.05073
[5] Catlin, P.A., Supereulerian graphs, a survey, J. graph theory, 16, 177-196, (1992) · Zbl 0771.05059
[6] Catlin, P.A.; Grossman, J.W.; Hobbs, A.M.; Lai, H.-J., Fractional arboricity, strength and principal partitions in graphs and matroids, Discrete appl. math., 40, 285-302, (1992) · Zbl 0773.05033
[7] Catlin, P.A.; Han, Z.Y.; Lai, H.-J., Graphs without spanning closed trails, J. discrete math., 160, 81-91, (1996) · Zbl 0859.05060
[8] Chen, Z.H.; Lai, H.-J., Reduction techniques for super-Eulerian graphs and related topics-a survey, (), 53-69
[9] Edmonds, J., Lehman’s switching game and a theorem of Tutte and Nash-Williams, J. res. nat. bur. stand sect. B, 69B, 73-77, (1965) · Zbl 0192.09102
[10] Erdös, P., Graph theory and probability, Can. J. math., 11, 34-38, (1959) · Zbl 0084.39602
[11] Hochstättler, Winfried; Jackson, Bill, Large circuits in binary matroids of large cogirth I, J. combin. theory, ser. B, 74, 35-52, (1998) · Zbl 0904.05023
[12] Jaeger, F., A note on Subeulerian graphs, J. graph theory, 3, 91-93, (1979)
[13] Kronk, H.V.; Mitchem, J., Critical point arboritic graphs, J. lond. math. soc., 9, 459-466, (1974/75) · Zbl 0298.05132
[14] Lai, H.-J.; Liu, B.; Liu, Y.; Shao, Y., Spanning cycles in regular matroids without \(M^\ast(K_5)\) minors, European J. combin., 29, 1, 298-310, (2008) · Zbl 1126.05031
[15] Nash-Williams, C.St.J.A., Edge-disjoint spanning trees of finite graphs, J. lond. math. soc., 36, 445-450, (1961) · Zbl 0102.38805
[16] Oxley, J.G., Matroid theory, (1992), Oxford University Press New York · Zbl 0784.05002
[17] Pulleyblank, W.R., A note on graphs spanned by Eulerian graphs, J. graph theory, 3, 309-310, (1979) · Zbl 0414.05040
[18] Seymour, P.D., Decomposition of regular matroids, J. combin. theory, ser. B, 28, 305-359, (1980) · Zbl 0443.05027
[19] Seymour, P.D., Matroids and multicommodity flows, European J. combin. theory ser. B, 2, 257-290, (1981) · Zbl 0479.05023
[20] Tutte, W.T., On the problem of decomposing a graph into \(n\) connected factors, J. lond. math. soc., 36, 221-230, (1961) · Zbl 0096.38001
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