Abstract
A graph g is k-ordered Hamiltonian, 2 ≤ k ≤ n, if for every ordered sequence S of k distinct vertices of G, there exists a Hamiltonian cycle that encounters S in the given order. In this article, we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least \(n+{{3k - 9} \over 2}\), then G is k-ordered Hamiltonian for \( k = 3,4, \cdots ,{\left\lfloor {\frac{n} {{19}}} \right\rfloor } \). We also show that the degree sum bound can be reduced to \( n + 2{\left\lfloor {\frac{k} {2}} \right\rfloor } - 2 \) if \( \kappa {\left( G \right)} \geqslant \frac{{3k - 1}} {2} \) or δ(G) ≥ 5k − 4. Several known results are generalized.
Similar content being viewed by others
References
Bondy, J.A., Murty, U.S.R. Graph theory with applications. Macmillan, London and Elsevier, New York, 1976
Faudree, J.R., Faudree, R.J., Gould, R.J., Jacobson, M.S., Lesniak, L. On k-ordered graphs. J. Graph Theory, 35: 69–82 (2000)
Kierstead, H.A., SÁrkÖzy, G.N., Selkow, S.M. On k-ordered Hamiltonian graphs. J. Graph Theory, 32: 17–25 (1999)
Ng, L., Schultz, M. k-ordered Hamiltonian graphs. J. Graph Theory, 24: 45–57 (1997)
Ore, O. A note on Hamiltonian circuits. Amer. Math. Monthly, 67: 55 (1960)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the National Natural Sciences Foundation of China (No. 19831080).
Rights and permissions
About this article
Cite this article
Hu, Zq., Tian, F. On k-ordered Graphs Involved Degree Sum. Acta Mathematicae Applicatae Sinica, English Series 19, 97–106 (2003). https://doi.org/10.1007/s10255-003-0085-3
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10255-003-0085-3