Proof of the Kalai-Meshulam conjecture

@article{Chudnovsky2018ProofOT,
  title={Proof of the Kalai-Meshulam conjecture},
  author={M. Chudnovsky and Alex D. Scott and Paul D. Seymour and Sophie Theresa Spirkl},
  journal={Israel Journal of Mathematics},
  year={2018},
  pages={1-23}
}
Let G be a graph, and let f G be the sum of (−1) ∣ A ∣ , over all stable sets A. If G is a cycle with length divisible by three, then f G = ±2. Motivated by topological considerations, G. Kalai and R. Meshulam [8] made the conjecture that, if no induced cycle of a graph G has length divisible by three, then ∣ f G ∣ ≤ 1. We prove this conjecture. 
On the topological Kalai-Meshulam conjecture
Chudnovsky, Scott, Seymour and Spirkl recently proved a conjecture by Kalai and Meshulam stating that the reduced Euler characteristic of the independence complex of a graph without induced cycles ofExpand
The Homotopy Type of the Independence Complex of Graphs with No Induced Cycles of Length Divisible by 3
We prove Engström’s conjecture that the independence complex of graphs with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. Our result strengthensExpand
THE HOMOTOPY TYPE OF THE INDEPENDENCE COMPLEX OF TERNARY GRAPHS
A graph G = (V,E) is said to be ternary if G contains no induced subgraph that is a cycle of length divisible by 3. A vertex set W ⊂ V is called an independent set of G if W does not contain an edgeExpand
Induced Subgraphs of Graphs With Large Chromatic Number. X. Holes of Specific Residue
TLDR
This paper unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. Expand
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 theExpand
A survey of $\chi$-boundedness
If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andras Gyarfas made a number of challenging conjectures about this in theExpand
The topology of solution spaces of combinatorial problems
Graph homomorphism is a notion almost as simple, notationally and conceptually, as graph coloring, but one that gives a rich mathematical structure, allowing for new fruitful connections with algebraExpand
Helly-type Problems
In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describeExpand
Detecting a long even hole
For each integer $\ell \geq 4$, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least $\ell$ and even
Detecting a Long Odd Hole
TLDR
This work gives a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least ℓ and odd. Expand
...
1
2
...

References

SHOWING 1-10 OF 21 REFERENCES
Induced Subgraphs of Graphs With Large Chromatic Number. X. Holes of Specific Residue
TLDR
This paper unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. Expand
Graphs with a Cycle of Length Divisible by Three
  • Guantao Chen, Akira Saito
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 1994
TLDR
It is proved that every graph G with minimum degree G contains a cycle of length divisible by three that was conjectured to be true by Barefoot, Clark, Douthett, and Entringer. Expand
Graphs with large chromatic number induce $3k$-cycles
TLDR
It is proved that graphs without induced cycles of length 3k have bounded chromatic number, which implies the very first case of a much broader question asserting that every graph with large Chromatic number induces a graph H such that the sum of the Betti numbers of the independence complex of H is also large. Expand
Algebraic Topology
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
On the independence complex of square grids
Abstract The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured byExpand
Hard squares with negative activity
We show that the hard-square lattice gas with activity z = −1 has a number of remarkable properties. We conjecture that all the eigenvalues of the transfer matrix are roots of unity. They fall intoExpand
Hard Squares with Negative Activity and Rhombus Tilings of the Plane
  • J. Jonsson
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 2006
TLDR
It is shown that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. Expand
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 theExpand
Graphs with no cycle length divisible by three
  • Ph. D. thesis,
  • 2017
When do a few colors suffice?”, https://gilkalai.wordpress.com/2014/12/19/when-afew-colors-suffice
  • 2014
...
1
2
3
...