Algorithms and Their Complexity

Bang‐Jensen, Jørgen; Gutin, Gregory

doi:10.1007/978-1-84800-998-1_18

Cited by 132 publications

(242 citation statements)

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“…Although some notion of connectivity in edge-colored graphs have already been known (see, e.g. Chapter 11 in [3]), in the absence of a counterpart to Menger's theorem and network flow theory, the task may seem daunting at first, perhaps even beyond reach. Yet the results are surprinsingly consistent with their counterparts from Graph Theory.…”

Section: Resultsmentioning

confidence: 99%

“…Properly colored paths and cycles have applications in various other fields, as in VLSI for compacting a programmable logical array [13]. Although a large body of work has already been done [3,4,5,6,8,16], in most of that previous work the number of colors was restricted to two. For instance, while it is well known that properly edge-colored hamiltonian cycles can be found efficiently in 2-edge colored complete graphs, it is a long standing question whether there exists a polynomial algorithm for finding such hamiltonian cycles in edge-colored complete graphs with three colors or more [6].…”

Section: Introduction and Notationmentioning

confidence: 99%

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Cycles and paths in edge‐colored graphs with given degrees

Abouelaoualim

Das

Véga

et al. 2009

Journal of Graph Theory

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Sufficient degree conditions for the existence of properly edge-colored cycles and paths in edge-colored graphs, multigraphs and random graphs are inverstigated. In particular, we prove that an edgecolored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to n+1 2 has properly edge-colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge-colored paths and hamiltonian paths between given vertices are considered as well.

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Section: Resultsmentioning

confidence: 99%

Section: Introduction and Notationmentioning

confidence: 99%

Cycles and paths in edge‐colored graphs with given degrees

Abouelaoualim

Das

Véga

et al. 2009

Journal of Graph Theory

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“…An oriented graph is a digraph without 2-cycles. For undefined digraph concepts we refer the reader to [3].…”

Section: Destroying Longest Cycles In Digraphsmentioning

confidence: 99%

Destroying longest cycles in graphs and digraphs

Aardt

Burger

Dunbar

et al. 2015

Discrete Applied Mathematics

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In 1978, C. Thomassen [Hypohamiltonian graphs and digraphs, Proceedings of the International Conference on the Theory and Applications of Graphs, Kalamazoo, 1976, Springer Verlag, pp. 557 -571] proved that in any graph one can destroy all the longest cycles by deleting at most one third of the vertices. We show that for graphs with circumference k ≤ 8 it suffices to remove at most 1/k of the vertices. The Petersen graph demonstrates that this result cannot be extended to include k = 9 but we show that in every graph with circumference nine we can destroy all 9-cycles by removing 1/5 * Corresponding author. of the vertices. We consider the analogous problem for digraphs and show that for digraphs with circumference k = 2, 3, it suffices to remove 1/k of the vertices. However this does not hold for k ≥ 4.

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“…, x p−1 x p } to T , we obtain a strong semi-complete digraph D (every pair of vertices has one or two edges between them). As Moon's theorem (a strong tournament is vertex-pancyclic) extends to strong semi-complete digraphs (see [1]), there exists an -cycle C of D through x. We will form a cycle C of T from C by using the same vertex set (in the same permutation).…”

Section: Progressmentioning

confidence: 99%

Vertex‐pancyclicity of hypertournaments

Yang

2009

Journal of Graph Theory

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DefinitionsLet V be a n-set (set of size n). Let E be the collection of all possible k-subsets (subsets of size k) of V , 2 ≤ k ≤ n, each taken in one of its k! possible permutations. A pair T = (V, E) is called a hypertournament, or a k-tournament. Each element of V is a vertex, and each ordered k-tuple of E is a hyperedge or, simply, an edge. For vertices u, v ∈ V and an edge e = (x 1 , . . . , x k ) ∈ E, we say u dominates v via edge e if u precedes v in e, that is if u = x i , v = x j , 1 ≤ i < j ≤ k. We denote this by uev. A path consists of an alternating sequencex 0 e 1 x 1 e 2 x 2 . . . x −1 e x of distinct vertices x i and distinct edges e i so that x i−1 dominates x i via e i , i = 1, . . . , . Such a path has length . A cycle is a path when all vertices are distinct except x 0 = x . A path (cycle) of T is Hamiltonian if it contains all vertices of T . A k-tournament T is pancyclic if it contains cycles of all possible lengths. It is vertex-pancyclic if each vertex of T is contained in cycles of all possible lengths. It is d-disjointvertex-pancyclic if each vertex of T is contained in d edge-disjointcycles for each possible length . A k-tournament T is strong if there is a path from u to v for each pair u, v ∈ V . The vertex set V and the edge collection E is also denoted V (T ) and E(T ), respectively. A hypertournament T is d-edge-connected if, for any two vertices u, v ∈ V , there are d pairwise edge-disjoint paths from u to v. For an ordinary graph (V, E), Γ(v) is the neighbors of v. For A ⊆ V , we denote by Γ(A) the set a∈A Γ(a). Known ResultsTheorem 1 (Redei). Every tournament has a Hamiltonian path.Theorem 2 (Moon). Every strong tournament is vertex-pancyclic.Gutin and Yeo [2] proved the following.

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Algorithms and Their Complexity

Cited by 132 publications

References 0 publications

Cycles and paths in edge‐colored graphs with given degrees

Cycles and paths in edge‐colored graphs with given degrees

Destroying longest cycles in graphs and digraphs

Vertex‐pancyclicity of hypertournaments

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