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Short Proofs of the Kneser-Lovász Coloring Principle [article]

James Aisenberg, Maria Luisa Bonet, Sam Buss, Adrian Crãciun, Gabriel Istrate
2015 arXiv   pre-print
We present a new counting-based combinatorial proof of the Kneser-Lov\'asz theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k.  ...  We prove that the propositional translations of the Kneser-Lov\'asz theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs.  ...  The extended Frege proof defines the instance of the Kneser-Lovasz principle Kneser n−1 k by discarding one node and one color.  ... 
arXiv:1505.05531v1 fatcat:ihz4qfdbwrehrit6eizx5okvv4

Short Proofs of the Kneser-Lovász Coloring Principle [chapter]

James Aisenberg, Maria Luisa Bonet, Sam Buss, Adrian Crãciun, Gabriel Istrate
2015 Lecture Notes in Computer Science  
We introduce new "truncated Tucker lemma" principles, which are miniaturizations of the octahedral Tucker lemma. The truncated Tucker lemma implies the Kneser-Lovász theorem.  ...  We present a new counting-based combinatorial proof of the Kneser-Lovász theorem that avoids the topological arguments of prior proofs for all but finitely many base cases.  ...  The extended Frege proof defines an instance of the Kneser-Lovász principle Kneser n−1 k by discarding one node and one color.  ... 
doi:10.1007/978-3-662-47666-6_4 fatcat:oi4nu2bcc5bxfgbem6g4bqplz4

Propositional Proofs in Frege and Extended Frege Systems (Abstract) [chapter]

Sam Buss
2015 Lecture Notes in Computer Science  
We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász  ...  We next state the results about the Kneser-Lovász principle. It is not hard to show that the (n, k)-Kneser graph can be colored with n−2k+2 colors.  ...  The (n, k)-Kneser graph cannot be colored with n−2k+1 colors.Note that the k = 1 case of the Theorem 4 is just the usual pigeonhole principle.It is straightforward to translate the Kneser-Lovász principle  ... 
doi:10.1007/978-3-319-20297-6_1 fatcat:vjlmno4i3bc3tgdds37zgc7f7m

Topological lower bounds for the chromatic number: A hierarchy [article]

Jiri Matousek, Günter M. Ziegler
2003 arXiv   pre-print
Such a lower bound was first introduced by Lovász in 1978, in his famous proof of the Kneser conjecture via Algebraic Topology.  ...  This conjecture stated that the Kneser graph _m,n, the graph with all k-element subsets of {1,2,...,n} as vertices and all pairs of disjoint sets as edges, has chromatic number n-2k+2.  ...  We also thank Wojchiech Chachólski and Péter Csorba for suggestions of examples with ind K = ind susp K.  ... 
arXiv:math/0208072v3 fatcat:o5p4zc7hzfdexn22t64t77a5ve

On the number of star-shaped classes in optimal colorings of Kneser graphs [article]

Hamid Reza Daneshpajouh
2022 arXiv   pre-print
The main aim of this paper is to provide a negative answer to the following question raised by James Aisenberg et al [Short proofs of the kneser-Lovasz coloring principle, Information and Computation,  ...  A family of sets is called star-shaped if all the members of the family have a point in common.  ...  Also a part of this work was done when the author was at the School of Mathematics of IPM as a guest researcher.  ... 
arXiv:2201.05605v2 fatcat:5ezb2slqizbb3ie6kbdixxmnfi

Intersection patterns of finite sets and of convex sets

Florian Frick
2016 arXiv   pre-print
We obtain an essentially elementary proof of the result of Schrijver on the chromatic number of stable Kneser graphs.  ...  As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture.  ...  Here we first present a short proof of Kneser's conjecture, which we will extend to the hypergraph setting in Section 4.  ... 
arXiv:1607.01003v1 fatcat:r7l6zmwwj5fuvjuvtg5ozdqgte

Sharp bounds for the chromatic number of random Kneser graphs [article]

Sergei Kiselev, Andrey Kupavskii
2021 arXiv   pre-print
One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of KG_n,k is equal to n-2k+2.  ...  Given positive integers n≥ 2k, the Kneser graph KG_n,k is a graph whose vertex set is the collection of all k-element subsets of the set {1,..., n}, with edges connecting pairs of disjoint sets.  ...  Florian pointed out the connection to Sarkaria's inequality.  ... 
arXiv:1810.01161v3 fatcat:qitoikqxerhitpglptqrvixtwu

Page 7152 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
Chapter 3 contains some famous amazing applications, like the ham sandwich theorem, necklace theorem, Lovasz-Kneser theorem and some other bounds on the chromatic number of Kneser graphs.  ...  In Chapter 2 several proofs of various versions of the Borsuk-Ulam theorem are given, including a combinatorial Tucker’s lemma.  ... 

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory [article]

Marc Roth, Philip Wellnitz
2021 arXiv   pre-print
In particular, we can drop the condition of ℋ being minor-closed for F-colorable graphs.  ...  Our main result is a construction based on Kneser graphs that associates every problem P in #𝖶[1] with two classes of graphs ℋ and 𝒢 such that the problem P is equivalent to the problem # HOM(ℋ→𝒢) of  ...  Acknowledgements We thank Karl Bringmann and Holger Dell for fruitful discussions and valuable feedback on early drafts of this work.  ... 
arXiv:1907.03850v2 fatcat:rg3upoulmve7jlsnloszbm22zu

Hypergraphs with many Kneser colorings

Carlos Hoppen, Yoshiharu Kohayakawa, Hanno Lefmann
2012 European journal of combinatorics (Print)  
It will be evident in the proof of Theorem 1.4 that the quest for the asymptotic value of KC(n, r, k, ℓ)  ...  The (C, r)-complete hypergraph H C ,r (n) is the hypergraph with vertex set [n] whose hyperedges are all the r-subsets of [n] containing some element of C as a subset.  ...  The authors are grateful to NUMEC/USP, Núcleo de Modelagem Estocástica e Complexidade of the University of São Paulo, for its hospitality.  ... 
doi:10.1016/j.ejc.2011.09.025 fatcat:yree3m2dsnbabl6je4t5xgc5yi

Coloring curves on surfaces [article]

Jonah Gaster, Joshua Evan Greene, Nicholas G. Vlamis
2016 arXiv   pre-print
We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes its clique number.  ...  Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry.  ...  Kneser exhibited a proper coloring of KG(n, k) using n − 2k + 2 colors, and Lovász proved its optimality by defining the neighborhood complex N (G) of a graph G, showing that the connectivity of N (G)  ... 
arXiv:1608.01589v1 fatcat:ssu76eujozgsxgnlmeui3pahoa

Parallel transport of Hom-complexes and the Lovasz conjecture [article]

Rade T. Zivaljevic
2005 arXiv   pre-print
The groupoid of projectivities, introduced by M. Joswig, serves as a basis for a construction of parallel transport of graph and more general Hom-complexes.  ...  In this framework we develop a general conceptual approach to the Lovasz Hom-conjecture, recently resolved by E. Babson and D. Kozlov, and extend their result from graphs to simplicial complexes.  ...  The Lovász conjecture One of central themes in topological combinatorics, after the landmark paper of Laszlo Lovász [21] where he proved the classical Kneser conjecture, has been the study and applications  ... 
arXiv:math/0506075v1 fatcat:zkbt5wng5nb35lnjxa576c7dra

Expected Chromatic Number of Random Subgraphs [article]

Ross Berkowitz, Pat Devlin, Catherine Lee, Henry Reichard, David Townley
2018 arXiv   pre-print
We also propose the stronger conjecture that for any fixed p ≤ 1/2, among all graphs of fixed chromatic number, E[χ(G_p)] is minimized by the complete graph.  ...  Given a graph G and p ∈ [0,1], let G_p denote the random subgraph of G obtained by keeping each edge independently with probability p.  ...  With this, we see that for sufficiently large k, the Kneser graphs KG 3k,k provide an infinite family of counterexamples to Question 2. A Appendix: Proof of Proposition 2 Proof of Proposition 2.  ... 
arXiv:1811.02018v1 fatcat:ysr4fquebzbxdfhte4g52bfkye

Hypergraphs with many Kneser colorings (Extended Version) [article]

Carlos Hoppen, Yoshiharu Kohayakawa, Hanno Lefmann
2011 arXiv   pre-print
For fixed positive integers r, k and ℓ with 1 ≤ℓ < r and an r-uniform hypergraph H, let κ (H, k,ℓ) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same  ...  This variant of a problem of Erdős and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdős--Ko--Rado Theorem on intersecting systems of sets [Intersection  ...  A hypergraph admitting a (k, ℓ)-Kneser coloring is called (k, ℓ)-Kneser colorable ((k, ℓ)-colorable, for short), and the number of (k, ℓ)-Kneser colorings of a hypergraph H is denoted by κ(H, k, ℓ).  ... 
arXiv:1102.5543v1 fatcat:7ljn22gyczcezjqa4rm4bo3ymm

COLORING CURVES ON SURFACES

JONAH GASTER, JOSHUA EVAN GREENE, NICHOLAS G. VLAMIS
2018 Forum of Mathematics, Sigma  
We also show that the graph of curves that represent a fixed nonzero homology class is uniquely $t$ -colorable, where $t$ denotes its clique number.  ...  Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry.  ...  We especially thank Ian for explaining how to use train tracks in place of hyperbolic geometry in the proof of Theorem 1.3. We also thank the referees for their thorough reviews.  ... 
doi:10.1017/fms.2018.12 fatcat:hbced6pt7zhi7l2waarnweapte
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