A proof of the Erdős-Faber-Lovász conjecture

Kang, Donghoon; Kelly, Tom; Kühn, Daniela; Methuku, Abhishek; Osthus, Deryk

doi:10.48550/arxiv.2101.04698

Cited by 5 publications

(30 citation statements)

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“…It remains to prove the bound f (m, G χ n ) ≤ m + n − 2 when m + n is sufficiently large. First we note the following immediate consequence of the main result in [13] (namely that the Erdős-Faber-Lovász conjecture holds for all sufficiently large n). Since χ(G 1 ) ≤ n, there is a partition I 1 , .…”

Section: Proof Of Theorem 14mentioning

confidence: 88%

“…An intermediate result of Kahn [10,Theorem 3] was also crucial to proving the Erdős-Faber-Lovász conjecture asymptotically. This conjecture states that a nearly disjoint union of n complete graphs, each on at most n vertices, has chromatic number at most n, and it was recently confirmed for all large n by Kang, Methuku, and the authors [13]. Note that the line graph of a k-bounded linear hypergraph is a union of nearly disjoint complete graphs, such that at most k of them contain any given vertex.…”

Section: Introductionmentioning

confidence: 87%

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A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

Kelly¹,

Kühn²,

Osthus³

2021

Preprint

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A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if G1, . . . , Gm are nearly disjoint graphs of maximum degree at most D, then the following holds. For every fixed C, if each vertex v ∈ m i=1 V (Gi) is contained in at most C of the graphs G1, . . . , Gm, then the (list) chromatic number of m i=1 Gi is at most D + o(D). This result confirms a special case of a conjecture of Vu and generalizes Kahn's bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy, and we derive this result from a more general list coloring result in the setting of 'color degrees' that also implies a result of Reed and Sudakov.

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Section: Proof Of Theorem 14mentioning

confidence: 88%

Section: Introductionmentioning

confidence: 87%

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

Kelly¹,

Kühn²,

Osthus³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…To construct H and ψ we iteratively apply the Rödl nibble to (the leftover of) H \ R to successively construct large matchings N i which are then removed from H \ R and form part of the colour classes of ψ. (The Rödl nibble is applied implicitly via [42,Corollary 4.3], which guarantees a large matching in a suitable hypergraph.) Crucially, each matching N i exhibits pseudorandom properties, which allow us to use some edges of R to extend N i into a matching M i (which will form a colour class of ψ) with nearly-perfect coverage of U , as desired.…”

Section: Is Covered By All But At Most One Of the Colour Classes Of ψ...mentioning

confidence: 99%

A proof of the Erdös-Faber-Lovász conjecture: Algorithmic aspects

Kang

Kelly

Kühn

et al. 2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

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“…Roughly speaking, the idea is that we also want to be able to claim that the obtained conflict-free matching M behaves as one would expect by considering probabilistic heuristics. In the usual setting, without a conflict system, such a tool was provided in [9] and has already found a number of applications (see for example [24,27]). Inevitably, this additional feature adds in technicality and length to our proof, but we believe it could be essential for future applications.…”

Section: Introductionmentioning

confidence: 99%

Conflict-free hypergraph matchings

Glock¹,

Joos²,

Kim³

et al. 2022

Preprint

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A celebrated theorem of Pippenger, and Frankl and Rödl states that every almostregular, uniform hypergraph H with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a conflict-free matching, where conflicts are encoded via a collection C of subsets C ⊆ E(H). We say that a matching M ⊆ E(H) is conflict-free if M does not contain an element of C as a subset. Under natural assumptions on C, we prove that H has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called "high-girth" Steiner systems. Our main tool is a random greedy algorithm which we call the "conflict-free matching process".

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A proof of the Erdős-Faber-Lovász conjecture

Cited by 5 publications

References 0 publications

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

A proof of the Erdös-Faber-Lovász conjecture: Algorithmic aspects

Conflict-free hypergraph matchings

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