An improved bound for the strong chromatic number

Haxell, Penny

doi:10.1002/jgt.20300

Cited by 18 publications

(22 citation statements)

References 13 publications

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“…Here, we prove a result that is asymptotically optimal for large s, except for the lower-order O( √ s) term: we show (algorithmically) that L(s; n, n) ≥ (s − O( √ s)) · n. An interesting fact is that this was not known even non-constructively before: Theorem 1.1 roughly gives L(s; n, n) ≥ (s/e) · n. We also give faster serial and perhaps the first RNC algorithms with good bounds, for the strong chromatic number. Strong coloring is quite well-studied [5,8,16,23,24], and is in turn useful in covering a matrix with Latin transversals [7].…”

Section: Introductionmentioning

confidence: 99%

A constructive algorithm for the Lovász Local Lemma on permutations

Harris¹,

Srinivasan²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations: the "lopsided" version of the LLL is usually at play here, and we do not yet have subexponential-time algorithms. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin Transversals: the best-known general result here (Bissacot et al., improving on Erdős and Spencer), states that any n × n matrix in which each entry appears at most (27/256)n times, has a Latin transversal. We present the first polynomial-time algorithm to construct such a transversal. Our approach also yields RNC algorithms: for Latin transversals, as well as the first efficient ones for the strong chromatic number and (special cases of) acyclic edge-coloring.

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Section: Introductionmentioning

confidence: 99%

A constructive algorithm for the Lovász Local Lemma on permutations

Harris¹,

Srinivasan²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In [22], it is shown that when b ≥ (11/4)∆ + Ω(1), such a coloring exists; this is the best bound currently known. Furthermore, the constant 11/4 cannot be improved to any number strictly less than 2.…”

Section: Strong Chromatic Numbermentioning

confidence: 98%

“…Strong coloring is quite well studied [5,8,14,21,22], and is in turn useful in covering a matrix with Latin transversals [7].…”

Section: Applicationsmentioning

confidence: 99%

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Harris¹,

Srinivasan²

2017

Theory of Comput.

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“…The best known general bound for the strong chromatic number of graphs G in terms of their maximum degree (G) is sχ(G) 3 (G) − 1, proved in [39]. (See also [40] for an asymptotically better bound.) Aharoni, Berger and Ziv [4] gave a nice simplification of the proof in [39], that gives the bound sχ(G) 3 (G).…”

Section: Strong Colouringmentioning

confidence: 99%

Finding independent transversals efficiently

Graf

Haxell

2020

Combinator. Probab. Comp.

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We give an efficient algorithm that, given a graph G and a partition V1,…,V m of its vertex set, finds either an independent transversal (an independent set {v1,…,v m } in G such that ${v_i} \in {V_i}$ for each i), or a subset ${\cal B}$ of vertex classes such that the subgraph of G induced by $\bigcup\nolimits_{\cal B}$ has a small dominating set. A non-algorithmic proof of this result has been known for a number of years and has been used to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.

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An improved bound for the strong chromatic number

Cited by 18 publications

References 13 publications

A constructive algorithm for the Lovász Local Lemma on permutations

A constructive algorithm for the Lovász Local Lemma on permutations

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Finding independent transversals efficiently

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