# Improved Sufficient Conditions for Hamiltonian Properties

Jens-P. Bode; Anika Fricke; Arnfried Kemnitz

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 2, page 329-334
- ISSN: 2083-5892

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topJens-P. Bode, Anika Fricke, and Arnfried Kemnitz. "Improved Sufficient Conditions for Hamiltonian Properties." Discussiones Mathematicae Graph Theory 35.2 (2015): 329-334. <http://eudml.org/doc/271094>.

@article{Jens2015,

abstract = {In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+ 1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity condition.},

author = {Jens-P. Bode, Anika Fricke, Arnfried Kemnitz},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {Hamiltonian; traceable; Hamiltonian-connected; Hamiltonian cycle; traceable graphs; Hamiltonian-connected graphs},

language = {eng},

number = {2},

pages = {329-334},

title = {Improved Sufficient Conditions for Hamiltonian Properties},

url = {http://eudml.org/doc/271094},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Jens-P. Bode

AU - Anika Fricke

AU - Arnfried Kemnitz

TI - Improved Sufficient Conditions for Hamiltonian Properties

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 2

SP - 329

EP - 334

AB - In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+ 1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity condition.

LA - eng

KW - Hamiltonian; traceable; Hamiltonian-connected; Hamiltonian cycle; traceable graphs; Hamiltonian-connected graphs

UR - http://eudml.org/doc/271094

ER -

## References

top- [1] J.-P. Bode, A. Kemnitz, I. Schiermeyer and A. Schwarz, Generalizing Bondy’s theorems on sufficient conditions for Hamiltonian properties, Congr. Numer. 203 (2010) 5-13. Zbl1229.05191
- [2] J.A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16 (Department of Combinatorics and Optimization, Faculty of Mathe- matics, University of Waterloo, Waterloo, Ontario, Canada, 1980).
- [3] J.A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15 (1976) 111-135. doi:10.1016/0012-365X(76)90078-9[Crossref] Zbl0331.05138
- [4] V. Chvátal and P. Erdős, A note on Hamiltonian circuits, Discrete Math. 2 (1972) 111-113. doi:10.1016/0012-365X(72)90079-9[Crossref]
- [5] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. s3-2 (1952) 69-81. doi:10.1112/plms/s3-2.1.69[Crossref] Zbl0047.17001
- [6] P. Fraisse, D≥-cycles and their applications for Hamiltonian graphs (LRI, Rapport de Recherche 276, Centre d’Orsay, Université de Paris-Sud, 1986).
- [7] O. Ore, Note on Hamiltonian circuits, Amer. Math. Monthly 67 (1960) 55.
- [8] O. Ore, Hamilton connected graphs, J. Math. Pures Appl. 42 (1963) 21-27. Zbl0106.37103

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