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Short Proofs of the Kneser-Lovász Coloring Principle
[article]
2015
arXiv
pre-print
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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]
2015
Lecture Notes in Computer Science
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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]
2015
Lecture Notes in Computer Science
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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]
2003
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
2016
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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
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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]
2021
arXiv
pre-print
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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
2012
European journal of combinatorics (Print)
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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]
2016
arXiv
pre-print
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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]
2005
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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]
2011
arXiv
pre-print
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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
2018
Forum of Mathematics, Sigma
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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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