Discussiones Mathematicae Graph Theory 27(1) (2007) 179-191
doi: 10.7151/dmgt.1354

[BIBTex] [PDF] [PS]

CYCLES THROUGH SPECIFIED VERTICES IN TRIANGLE-FREE GRAPHS

Daniel Paulusma

Department of Computer Science, Durham University
Science Laboratories, South Road, Durham DH1 3LE, England
e-mail: daniel.paulusma@durham.ac.uk

Kiyoshi Yoshimoto

Department of Mathematics
College of Science and Technology
Nihon University, Tokyo 101-8308, Japan
e-mail: yosimoto@math.cst.nihon-u.ac.jp

Abstract

Let G be a triangle-free graph with δ(G) ≥ 2 and σ4(G) ≥ |V(G)|+2. Let S ⊂ V(G) consist of less than σ4/4+ 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ4 are best possible.

Keywords: cycle, path, triangle-free graph.

2000 Mathematics Subject Classification: 05C38, 05C45.

References

[1] P. Ash and B. Jackson, Dominating cycles in bipartite graphs, in: Progress in Graph Theory, J.A. Bondy, U.S.R. Murty, eds., (Academic Press, 1984), 81-87.
[2] B. Bollobás and G. Brightwell, Cycles through specified vertices, Combinatorica 13 (1993) 137-155.
[3] J.A. Bondy, Longest Paths and Cycles in Graphs of High Degree, Research Report CORR 80-16 (1980).
[4] J.A. Bondy and L. Lovász, Cycles through specified vertices of a graph, Combinatorica 1 (1981) 117-140, doi: 10.1007/BF02579268.
[5] H. Broersma, H. Li, J. Li, F. Tian and H.J. Veldman, Cycles through subsets with large degree sums, Discrete Math. 171 (1997) 43-54, doi: 10.1016/S0012-365X(96)00071-4.
[6] R. Diestel, Graph Theory, Second edition, Graduate Texts in Mathematics 173, Springer (2000).
[7] Y. Egawa, R. Glas and S.C. Locke, Cycles and paths trough specified vertices in k-connected graphs, J. Combin. Theory (B) 52 (1991) 20-29, doi: 10.1016/0095-8956(91)90086-Y.
[8] J. Harant, On paths and cycles through specified vertices, Discrete Math. 286 (2004) 95-98, doi: 10.1016/j.disc.2003.11.059.
[9] H. Enomoto, J. van den Heuvel, A. Kaneko and A. Saito, Relative length of long paths and cycles in graphs with large degree sums, J. Graph Theory 20 (1995) 213-225, doi: 10.1002/jgt.3190200210.
[10] D.A. Holton, Cycles through specified vertices in k-connected regular graphs, Ars Combin. 13 (1982) 129-143.
[11] O. Ore, Note on hamiltonian circuits, American Mathematical Monthly 67 (1960) 55, doi: 10.2307/2308928.
[12] K. Ota, Cycles through prescribed vertices with large degree sum, Discrete Math. 145 (1995) 201-210, doi: 10.1016/0012-365X(94)00036-I.
[13] D. Paulusma and K. Yoshimoto, Relative length of longest paths and longest cycles in triangle-free graphs, submitted,
http://www.math.cst.nihon-u.ac.jp/~yosimoto/paper/related_length1_sub.pdf.
[14] A. Saito, Long cycles through specified vertices in a graph, J. Combin. Theory (B) 47 (1989) 220-230, doi: 10.1016/0095-8956(89)90021-X.
[15] L. Stacho, Cycles through specified vertices in 1-tough graphs, Ars Combin. 56 (2000) 263-269.
[16] K. Yoshimoto, Edge degree conditions and all longest cycles which are dominating, submitted.
[17] S.J. Zheng, Cycles and paths through specified vertices, Journal of Nanjing Normal University, Natural Science Edition, Nanjing Shida Xuebao, Ziran Kexue Ban 23 (2000) 9-13.

Received 8 March 2006
Revised 31 October 2006