Abstract
Consider any undirected and simple graph G = (V,E), where V and E denote the vertex set and the edge set of G, respectively. Let |G| = |V| = n ≥ 3. The well-known Ore’s theorem states that if deg G (u) + deg G (v) ≥ n holds for each pair of nonadjacent vertices u and v of G, then G is hamiltonian. A similar theorem given by Erdös is as follows: if deg G (u) + deg G (v) ≥ n + 1 holds for each pair of nonadjacent vertices u and v of G, then G is hamiltonian-connected. In this paper, we improve both theorems by showing that any graph G satisfying the condition in Ore’s theorem is hamiltonian-connected unless G belongs to two exceptional families.
This research was partially supported by the National Science Council of the Republic of China under contract NSC 101-2115-M-033-003-.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bondy, J.A., Mutry, U.S.R.: Graph Theory with Applications. North-Holland, New York (1980)
Erdös, P., Gallai, T.: On Maximal Paths and Circuits of Graphs. Acta Mathematica Hungarica 10, 337–356 (1959)
Kao, S.S., Hsu, K.M., Hsu, L.H.: Cubic planar hamiltonian graphs of various types. Discrete Mathematics 309, 1364–1389 (2006)
Ore, O.: Note on Hamilton Circuit. American Mathematical Monthly 67, 55 (1960)
Su, H., Shih, Y.-K., Kao, S.-S.: On the 1-fault hamiltonicity for graphs satisfying Ore’s theorem. Information Processing Letters 112, 839–843 (2012)
Zhao, K., Chen, D., Lin, Y., Li, Z., Zeng, K.: Some new sufficient conditions and Hamiltonian-connected graphs. Procedia Engineerig 24, 278–281 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shih, YK., Su, H., Kao, SS. (2013). On the Hamiltonian-Connectedness for Graphs Satisfying Ore’s Theorem. In: Chang, RS., Jain, L., Peng, SL. (eds) Advances in Intelligent Systems and Applications - Volume 1. Smart Innovation, Systems and Technologies, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35452-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-35452-6_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35451-9
Online ISBN: 978-3-642-35452-6
eBook Packages: EngineeringEngineering (R0)