Hamiltonian problems in edge-colored complete graphs and eulerian cycles in edge-colored graphs : some complexity results

Benkouar, A.; Manoussakis, Yannis; Paschos, Vangélis Th.; Saad, Rachid

doi:10.1051/ro/1996300404171

Cited by 22 publications

(25 citation statements)

References 10 publications

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“…However, a proper Hamiltonian path can be determined in polynomial time in c-edge-colored complete graphs for c ≥ 2 [8]. It is also polynomial to find a proper Hamiltonian cycle in c-edgecolored complete graphs for c = 2, see [5], but it is still open to determine the computational complexity for c ≥ 3 [6]. Many other results for edge-colored multigraphs can be found in the survey by Bang-Jensen and Gutin [3].…”

Section: Introductionmentioning

confidence: 99%

Proper Hamiltonian cycles in edge-colored multigraphs

Águeda

Borozan

Díaz

et al. 2017

Discrete Mathematics

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A c-edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.

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Section: Introductionmentioning

confidence: 99%

Proper Hamiltonian cycles in edge-colored multigraphs

Águeda

Borozan

Díaz

et al. 2017

Discrete Mathematics

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“…However it is polynomial to find a proper Hamiltonian path in c-edgecoloured complete graphs, c ≥ 2 [7]. It is also polynomial to find a proper Hamiltonian cycle in 2-edge-coloured complete graphs [4], but it is still open to determine the computational complexity for c ≥ 3 [5]. Many other results for edge-coloured multigraphs can be found in the survey by Bang-Jensen and Gutin [2].…”

Section: Introductionmentioning

confidence: 99%

Proper Hamiltonian Paths in Edge-Coloured Multigraphs

Águeda

Borozan

Groshaus

et al. 2017

Graphs and Combinatorics

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Given a c-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an edge-coloured multigraph to guarantee the existence of a proper Hamiltonian path, involving various parameters as the number of edges, the number of colours, the rainbow degree and the connectivity.

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“…In an important application, Pevzner [24] used compatible circuits in undirected Eulerian graphs to reconstruct DNA from its segments. Benkouar et al [3] gave a polynomial time algorithm for finding a compatible circuit in a colored Eulerian undirected graph, providing an alternate proof of Kotzig's theorem. They claimed that a similar algorithm holds for digraphs and gave a statement characterizing the existence of a compatible circuit in a colored Eulerian digraph.…”

Section: Introductionmentioning

confidence: 99%

Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

Carraher¹,

Hartke²

2013

SIAM J. Discrete Math.

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Let G be an Eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of G is an Eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compatible circuits in edge-colored Eulerian undirected graphs. From our characterization for digraphs we develop a polynomial time algorithm that determines the existence of a compatible circuit in an edge-colored Eulerian digraph and produces a compatible circuit if one exists. Our results use the fact that rainbow spanning trees have been characterized in edge-colored undirected multigraphs. We provide another graph-theoretical proof of this fact.

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Hamiltonian problems in edge-colored complete graphs and eulerian cycles in edge-colored graphs : some complexity results

Cited by 22 publications

References 10 publications

Proper Hamiltonian cycles in edge-colored multigraphs

Proper Hamiltonian cycles in edge-colored multigraphs

Proper Hamiltonian Paths in Edge-Coloured Multigraphs

Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

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