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International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 262-273 | Cite as

Some Hamiltonian Properties of One-Conflict Graphs

  • Christian Laforest
  • Benjamin Momège
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

Dirac’s and Ore’s conditions (1952 and 1960) are well known and classical sufficient conditions for a graph to contain a Hamiltonian cycle and they are generalized in 1976 by the Bondy-Chvátal Theorem. In this paper, we add constraints, called conflicts. A conflict in a graph G is a pair of distinct edges of G. We denote by \((G,\mathcal {C}onf)\) a graph G with a set of conflicts \(\mathcal {C}onf\). A path without conflict P in \((G,\mathcal {C}onf)\) is a path P in G such that for any edges \(e,e'\) of P, \(\{e,e'\}\notin \mathcal {C}onf\). In this paper we consider graph with conflicts such that each vertex is not incident to the edges of more than one conflict. We call such graphs one-conflict graphs. We present sufficient conditions for one-conflict graphs to have a Hamiltonian path or cycle without conflict.

Keywords

Graph Conflict Hamiltonian Path Cycle 

Notes

Acknowledgements

We thank Mamadou M. Kanté and anonymous referees for reading and helping to improve a first version of this work.

References

  1. 1.
    Bondy, J.A., Chvátal, V.: A method in graph theory. Discrete Math. 15, 111–135 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, London Ltd (2010)Google Scholar
  3. 3.
    Dirac, G.A.: Some theorems on abstract graphs. Proc. London Math. Soc. 2, 69–81 (1952)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Dvořák, Z.: Two-factors in orientated graphs with forbidden transitions. Discrete Math. 309(1), 104–112 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kanté, M.M., Laforest, C., Momège, B.: An exact algorithm to check the existence of (elementary) paths and a generalisation of the cut problem in graphs with forbidden transitions. In: van Emde Boas, P., Groen, F.C.A., Italiano, G.F., Nawrocki, J., Sack, H. (eds.) SOFSEM 2013. LNCS, vol. 7741, pp. 257–267. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  6. 6.
    Kanté, M.M., Laforest, C., Momège, B.: Trees in graphs with conflict edges or forbidden transitions. In: Chan, T.-H.H., Lau, L.C., Trevisan, L. (eds.) TAMC 2013. LNCS, vol. 7876, pp. 343–354. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  7. 7.
    Li, H.: Generalizations of Dirac’s theorem in hamiltonian graph theory - a survey. Discrete Math. 313(19), 2034–2053 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ore, Ø.: Note on Hamiltonian circuits. Am. Math. Mon. 67, 55 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Szeider, S.: Finding paths in graphs avoiding forbidden transitions. Discrete Appl. Math. 126(2–3), 261–273 (2003)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LIMOSCNRS UMR 6158 – Université Blaise Pascal, Clermont-FerrandAubiére CedexFrance

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