Path and cycle sub-ramsey numbers and an edge-colouring conjecture

Hahn, Geňa; Thomassen, Carsten

doi:10.1016/0012-365x(86)90038-5

Cited by 71 publications

(58 citation statements)

References 2 publications

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“…Using inequalities (9) and (10), we deduce OP T (I ′ ) = OP T (I) + 2. Now, if T is a ρ-approximation for MinLTSP (2) , we deduce d(P ) ≤ ρ OP T (I) + 2(ρ − 1) ≤ (ρ + ε)OP T (I) when n is large enough.…”

Section: ⊓ ⊔mentioning

confidence: 97%

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On Labeled Traveling Salesman Problems

Couëtoux

Gourvès

Monnot

et al. 2008

Algorithms and Computation

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Abstract. We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approximation algorithms for both versions of the problem. For the maximization version we give a 1 2 -approximation algorithm and show that it is APXhard. For the minimization version, we show that it is not approximable within n 1−ǫ for every ǫ > 0. When every color appears in the graph at most r times and r is an increasing function of n the problem is notFor fixed constant r we analyze a polynomialtime (r + Hr)/2-approximation algorithm (Hr is the r-th harmonic number), and prove APX-hardness. Analysis of the studied algorithms is shown to be tight.

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Section: ⊓ ⊔mentioning

confidence: 97%

“…A great amount of work that followed concerned identification of such conditions and bounds on the number of colors [4,1,7,9]. The optimization problems that we consider here were shown to be NP-hard in [4].…”

Section: Introductionmentioning

confidence: 99%

On Labeled Traveling Salesman Problems

Couëtoux

Gourvès

Monnot

et al. 2008

Algorithms and Computation

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“…The case m = n is also commonly studied (and includes Latin squares as a special case), and we will also focus on THEORY OF COMPUTING, Volume 13 (17), 2017, pp. 1-41 these.…”

Section: Applicationsmentioning

confidence: 99%

“…Finally, THEORY OF COMPUTING, Volume 13 (17), 2017, pp. we conclude in Section 9 with a discussion of future goals for the construction of a generalized LLL algorithm.…”

Section: Outlinementioning

confidence: 99%

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Harris¹,

Srinivasan²

2017

Theory of Comput.

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“…This is of some interest in view of the remark in the last paragraph of Section 3. For a related result, the reader is referred to Frieze and Reed [13], where it is proved that, for some large constant A, any r-bounded edge-colouring of the complete graph K n admits a multicoloured Hamilton cycle, where r = r(n) = n/A log n. Hahn and Thomassen [14] conjecture that the statement above holds for some r = r(n) = Ω(n).…”

Section: Some Generalisationsmentioning

confidence: 99%

On an anti‐Ramsey property of Ramanujan graphs

Haxell

Kohayakawa

1995

Random Struct Algorithms

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Abstract. If G and H are graphs, we write G → H (respectively, G → TH) if for any proper edge-colouring γ of G there is a subgraph H ′ ⊂ G of G isomorphic to H (respectively, isomorphic to a subdivision of H) such that γ is injective on E(H ′ ). Let us write C ℓ for the cycle of length ℓ. Spencer (cf. Erdős [10]) asked whether for any g ≥ 3 there is a graph G = G g such that (i) G has girth g(G) at least g, and (ii) G → T C 3 . Recently, Rödl and Tuza [22] answered this question in the affirmative by proving, using non-constructive methods, a result that implies that for any t ≥ 1 there is a graph G = G t of girth t + 2 such that G → C 2t+2 . In particular, condition (ii) may be strengthened to (iii) G → C ℓ for some ℓ = ℓ(G). For G = G t above ℓ = ℓ(G) = 2t + 2 = 2g(G) − 2. Here, we show that suitable Ramanujan graphs constructed by Lubotzky, Phillips, and Sarnak [18] are explicit examples of graphs G = G g satisfying (i) and (iii) above. For such graphs ℓ = ℓ(G) in (iii) may be taken to be roughly equal to (3/2)g(G), thus considerably improving the value 2g(G) − 2 given in the result of Rödl and Tuza. It is not known whether there are graphs G of arbitrarily large girth for which (iii) holds with ℓ = ℓ(G) = g(G).

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Path and cycle sub-ramsey numbers and an edge-colouring conjecture

Cited by 71 publications

References 2 publications

On Labeled Traveling Salesman Problems

On Labeled Traveling Salesman Problems

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On an anti‐Ramsey property of Ramanujan graphs

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