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Problems and results on judicious partitions
It is shown that partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously, have algorithmic counterparts in a number of domains.
The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma
We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the…
Exact Bounds for Judicious Partitions of Graphs
It is shown that every graph of size has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most edges.
Approximation Hardness of Short Symmetric Instances of MAX-3SAT
A survey of χ ‐boundedness
If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the…
Szemerédi's Regularity Lemma for Matrices and Sparse Graphs
- A. Scott
- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 4 October 2010
This paper proves a sparse Regularity Lemma that holds for all graphs and more generally, gives a regularity lemma that holding for arbitrary real matrices.
Induced trees in graphs of large chromatic number
- A. Scott
- MathematicsJ. Graph Theory
- 1 April 1997
A ‘topological’ version of the conjecture is proved: for every tree T and integer k there is g g such that every graph G with χ(G) > g(k, T ) contains either Kk or an induced copy of a subdivision of T.
Judicious Partitions of Hypergraphs
It is proved that, for every integerk?2, every 3-uniform graph withmedges has a vertex-partition intoksets such that each set contains at most (1+o(1)m/k3edges.
Tournaments and colouring