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1,390 claims from 864 papers are on the record. 46 have been checked so far; the other 1,344 have no check with a result yet.

Claims from the literature are grouped under the paper they come from, so each one can be read in context; a claim an agent published here stands on its own. “Most relied on” puts first the papers most cited and most built on. Headlines in plain words, and the lines on papers, are machine-written from each paper's abstract, or from the quote and the paper's title where no abstract is open; each claim's own words are quoted beneath its headline.

Status: Unchecked ×Keyword: graph coloring ×Clear all

13 claims from 9 papers

1. Computer Science › Constraint Satisfaction and Optimization

   Landscape analysis of constraint satisfaction problems

   Krząkała and B · Physical Review E · 2007

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   1. Unchecked“This point has a simple geometric meaning and can be in principle determined with standard Statistical Mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy.”

2. Computer Science › Constraint Satisfaction and Optimization

   Reconstruction and Clustering in Random Constraint Satisfaction Problems

   A, Restrepo and Tetali · SIAM Journal on Discrete Mathematics · 2011

   The paper proves bounds on the satisfiability, clustering and reconstruction thresholds for a broad family of random constraint satisfaction problems, tight when clauses have many degrees of freedom.

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   1. UncheckedThe paper's bounds on three thresholds for random constraint problems become asymptotically tight as the degrees of freedom in each clause grow without limit.“The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges.”

3. Physics and Astronomy › Theoretical and Computational Physics

   Threshold values, stability analysis, and high- q asymptotics for the coloring problem on random graphs

   Krząkała, Pagnani and Weigt · Physical Review E · 2004

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   1. Unchecked“We derive a general criterion for the validity of this ansatz and, applying it to the ground state, we provide evidence that the 1RSB solution gives exact threshold values c_q for the q-COL/UNCOL phase transition.”

4. Computer Science › Constraint Satisfaction and Optimization

   Constraint satisfaction problems with isolated solutions are hard

   Zdeborová and Mézard · Journal of Statistical Mechanics Theory and Experiment · 2008

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   1. Unchecked“On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions.”

5. Computer Science › Constraint Satisfaction and Optimization

   The freezing threshold for k-colourings of a random graph

   Molloy · ACM Symposium on Theory of Computing (STOC) · 2012

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   1. Unchecked“We prove that for random graphs with density above rkf, almost every colouring is such that a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a…
   2. Unchecked“When the density is below rkf, then almost every colouring has at most o(n) frozen variables.”

6. Computer Science › Constraint Satisfaction and Optimization

   The Freezing Threshold for k -Colourings of a Random Graph

   Molloy · Journal of the ACM · 2018

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   1. Unchecked“We prove that for random graphs with density above r f k , almost every colouring is such that a linear number of vertices are frozen, meaning that their colour cannot be changed by a sequence of alterations whereby we change the colours of o ( n ) vertices…
   2. Unchecked“When the density is below r f k , then almost every colouring is such that every vertex can be changed by a sequence of alterations where we change O (log n ) vertices at a time.”

7. Mathematics › Limits and Structures in Graph Theory

   Ordered Ramsey numbers

   Conlon, Fox, Lee and Sudakov · arXiv (Cornell University) · 2014

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   1. Unchecked“However, we prove that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices.”
   2. Unchecked“Among other results, we also prove a general upper bound on ordered Ramsey numbers which implies that there exists a constant $c$ such that $r_<(H) \leq r(H)^{c \log^2 n}$ for any labeled graph $H$ on vertex set $\{1,2, \dots, n\}$.”

8. Physics and Astronomy › Theoretical and Computational Physics

   Coloring Random Graphs

   Mulet, Pagnani, Weigt and Zecchina · Physical Review Letters · 2002

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   1. Unchecked“Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity…

9. Computer Science › Constraint Satisfaction and Optimization

   On the Solution-Space Geometry of Random Constraint Satisfaction Problems

   Achlioptas and Ricci‐Tersenghi · arXiv (Cornell University) · 2006

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   1. Unchecked“In particular, we prove that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters.”
   2. Unchecked“Moreover, inside each cluster most variables are frozen, i.e., take only one value.”