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Rainbow fractional matchings [article]

Ron Aharoni, Ron Holzman, Zilin Jiang
2019 arXiv   pre-print
. , E_ rn of (not necessarily distinct) sets of edges in an r-uniform hypergraph, each having a fractional matching of size n, has a rainbow fractional matching of size n (that is, a set of edges from  ...  distinct E_i's which supports such a fractional matching).  ...  Our assumption that there is no rainbow fractional matching of size n means that M ⊆X.  ... 
arXiv:1805.09732v2 fatcat:lezlsdhkljhafl2d7sg3tvxbku

Rainbow spanning structures in graph and hypergraph systems [article]

Yangyang Cheng, Jie Han, Bin Wang, Guanghui Wang
2021 arXiv   pre-print
; (2) we prove that for 1≤ d≤ k-1, essentially a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems.  ...  Note that if |E(H)|=m, then a rainbow H consists of exactly one edge from each G_i. Our main results are on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions.  ...  −d contains a perfect fractional matching.  ... 
arXiv:2105.10219v1 fatcat:56jllyypbzdrxe3tduq2sl654q

Large rainbow matchings in general graphs [article]

Ron Aharoni, Eli Berger, Maria Chudnovsky, David Howard, Paul Seymour
2018 arXiv   pre-print
We prove that any 3n-2 matchings of size n have a partial rainbow matching of size n.  ...  By a theorem of Drisko, any 2n-1 matchings of size n in a bipartite graph have a partial rainbow matching of size n.  ...  fractional matching of size m.  ... 
arXiv:1611.03648v5 fatcat:l46vgyrob5g7vescglbo7c4atq

Envy-free two-player m-cake and three-player two-cake divisions

Nicolas Lebert, Frédéric Meunier, Quentin Carbonneaux
2013 Operations Research Letters  
The proof relies on a theorem by Gyárfás linking the matching number and the fractional matching number in m-partite hypergraphs. Date: May 3, 2013.  ...  It is easy to see that unfortunately such a bipartite graph does not contain necessarily a rainbow matching.  ...  We would try to find a rainbow matching in a bipartite graph with r vertices on each side, 2r − 1 distinct edges colored in 4 colors, r/2 edges of each color, and with a perfect matching.  ... 
doi:10.1016/j.orl.2013.07.010 fatcat:mrpe5bf56zbdrefocr37hz4oty

Integrality gaps for colorful matchings

Steven Kelk, Georgios Stamoulis
2019 Discrete Optimization  
We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem.  ...  To finish the proof, we will give a bound on the fractional dual solution value which, by duality theory, gives an upper bound on the fractional value (and hence integrality gap) for the fractional Rainbow  ...  The results of the previous section used the fact that the color bounds were fractional numbers so a very natural question is whether we can find instances for the BCM (or Rainbow Matching) problem with  ... 
doi:10.1016/j.disopt.2018.12.003 fatcat:64jqlwfopjfrxi4dczlirzqxma

On the rainbow matching conjecture for 3-uniform hypergraphs [article]

Jun Gao, Hongliang Lu, Jie Ma, Xingxing Yu
2021 arXiv   pre-print
We prove that there exists an absolute constant n_0 such that this rainbow version holds for k=3 and n≥ n_0. We convert this rainbow matching problem to a matching problem on a special hypergraph H.  ...  Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erdős matching conjecture: For positive integers n,k,m with n≥ km, if each of the families F_1,  ...  matching if and only if F admits a rainbow matching.  ... 
arXiv:2011.14363v3 fatcat:zlumywhf6bh5vgg73flrtwyrva

The rainbow trout genome provides novel insights into evolution after whole-genome duplication in vertebrates

Camille Berthelot, Frédéric Brunet, Domitille Chalopin, Amélie Juanchich, Maria Bernard, Benjamin Noël, Pascal Bento, Corinne Da Silva, Karine Labadie, Adriana Alberti, Jean-Marc Aury, Alexandra Louis (+18 others)
2014 Nature Communications  
(plus matches greater than 0.8X best matches) and each matched protein was then realigned using Genewise 40 on the same trout genomic region. 93% of these teleost proteins matched at 41,300 different  ...  Due to a relatively recent WGD, the rainbow trout thus provides a unique opportunity to better understand the early steps of gene fractionation.  ...  Material from the androgenetic doubled haploid rainbow trout line used for genome sequencing was provided by G.H.T. and from the gynogenetic doubled haploid line for transcriptome sequencing by E.Q.  ... 
doi:10.1038/ncomms4657 pmid:24755649 pmcid:PMC4071752 fatcat:45ciflc6s5an3e4o4milcz7nn4

The n-queens completion problem [article]

Stefan Glock, David Munhá Correia, Benny Sudakov
2022 arXiv   pre-print
Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.  ...  Hence, M is rainbow. It remains to prove that this almost-perfect matching can be turned into a perfect rainbow matching.  ...  In the first step, we find an approximate rainbow matching, that is, a rainbow matching which covers all but o(n) vertices.  ... 
arXiv:2111.11402v2 fatcat:gbgp3l4r2vfzzkqvrtaicnk2gi

On a Generalization of the Ryser-Brualdi-Stein Conjecture [article]

Ron Aharoni, Pierre Charbit, David Howard
2013 arXiv   pre-print
A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i.  ...  no rainbow matching: Example 1.5 (Wanless) .  ...  By the definition of f (r, k), the matchings F ′ i have a (full) rainbow matching, and removing the edges belonging to Q yields a partial rainbow matching of size at least k − p.  ... 
arXiv:1305.6164v1 fatcat:inqseltanjgdbbxrc27np5eqey

Transcriptome Assembly, Gene Annotation and Tissue Gene Expression Atlas of the Rainbow Trout

Mohamed Salem, Bam Paneru, Rafet Al-Tobasei, Fatima Abdouni, Gary H. Thorgaard, Caird E. Rexroad, Jianbo Yao, Pankaj Jaiswal
2015 PLoS ONE  
A total of 4,146 contigs (9.2%), including 710 full-length sequences, did not match any mRNA sequences in the current rainbow trout genome reference.  ...  In this study, nonnormalized cDNA libraries were sequenced from 13 different tissues of a single doubled haploid rainbow trout from the same source used for the rainbow trout genome sequence.  ...  On the other hand, 12,207 (9.3%) marine rainbow trout transcripts did not match any of the Trinity contigs.  ... 
doi:10.1371/journal.pone.0121778 pmid:25793877 pmcid:PMC4368115 fatcat:kjfx75jk4fbfzmrovy6o7j5vgy

A rainbow blow-up lemma [article]

Stefan Glock, Felix Joos
2018 arXiv   pre-print
This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs.  ...  Kim, Kühn, Kupavskii and Osthus exploit this to prove several rainbow decomposition results.  ...  In contrast to this, by far not every perfect matching σ of A i+1 i yields a rainbow embedding of H[X 1 ∪ · · · ∪ X i+1 ].  ... 
arXiv:1802.07700v2 fatcat:ukbd42szwfapjdxkaxek45enoq

Choice Functions [article]

Ron Aharoni, Joseph Briggs
2021 arXiv   pre-print
This is a survey paper on rainbow sets (another name for "choice functions").  ...  The paper contains some new results: (1) theorems on scrambled versions, in which the sets are re-shuffled before choosing the rainbow set, and (2) results on weighted and cooperative versions - to be  ...  The fractional version of this conjecture was proved in [14] . It says that any 2n matchings of size n, in a general graph, have a rainbow set of edges possessing a fractional matching of size n.  ... 
arXiv:2107.12881v1 fatcat:jpeitu42ofe4hghqhkbqdddl3y

Page 559 of Endocrinology Vol. 60, Issue 4 [page]

1957 Endocrinology  
The diameter of this die was bored to match the internal diameter of a pellet gun. Rainbow trout varying in age from 6 to 19 months were employed.  ...  Fraction #2: To the supernatant fluid remaining after the precipitation of Fraction #1 at least one volume of acetone was added.  ... 

Improved Inapproximability of Rainbow Coloring [article]

Per Austrin, Amey Bhangale, Aditya Potukuchi
2018 arXiv   pre-print
Previously, this was only known for rainbow k/2 -colorable hypergraphs (Guruswami and Lee, SODA 2015).  ...  We prove that given a rainbow (k - 2√(k))-colorable k-uniform hypergraph, it is NP-hard to find a normal 2-coloring.  ...  But any x that matches such an w ′ also matches w and in particular it follows that w ∈ A j and hence A j = A j .  ... 
arXiv:1810.02784v3 fatcat:dh6lpftmarepng4kiv26m646xe

Choice Functions in the Intersection of Matroids

Joseph Briggs, Minki Kim
2019 Electronic Journal of Combinatorics  
We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d=r\lceil k\rceil -r+1$ functions of size (=sum  ...  of values) $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i), ~i \le d$, supporting a function with the same properties.  ...  Applying Theorem 6 then yields a fractional matching f whose support is rainbow with respect to the sets E j .  ... 
doi:10.37236/8844 fatcat:vuxxpfm7i5fzzcdk2lihdh2dnm
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