# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## 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.