International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 262-273

# 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.

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