# Lower bounds on the chromatic number of random graphs

@article{Ayre2021LowerBO, title={Lower bounds on the chromatic number of random graphs}, author={Peter J. Ayre and Amin Coja-Oghlan and Catherine S. Greenhill}, journal={ArXiv}, year={2021}, volume={abs/1812.09691} }

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborova and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we obtain a variational formula. As an application we calculate improved explicit lower bounds on…

## One Citation

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The replica symmetry breaking phase transition predicted by physicists is pinpointed, and it is proved that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound.

## References

SHOWING 1-10 OF 72 REFERENCES

Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

- Mathematics, Computer ScienceSTOC '10
- 2010

We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erdos-Renyi (ER) graph G(N,c/N) and random r-regular…

On the chromatic number of a random 5-regular graph

- Computer ScienceJ. Graph Theory
- 2009

It is shown that the chromatic number of a random 5-regular graph is asymptotically almost surely equal to 3, provided a certain four-variable function has a unique maximum at a given point in a bounded domain.

On the chromatic number of a random 5-regular graph

- Mathematics
- 2009

It was only recently shown by Shi and Wormald, using the differential equation method to analyze an appropriate algorithm, that a random 5-regular graph asymptotically almost surely has chromatic…

On the chromatic number of random d-regular graphs

- Mathematics
- 2008

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d (2k−3)log(k−1), then the…

COMBINATORIAL APPROACH TO THE INTERPOLATION METHOD AND SCALING LIMITS IN SPARSE RANDOM GRAPHS

- Mathematics, Physics
- 2009

We establish the existence of free energy limits for several combinatorial models on Erd\"{o}s-R\'{e}nyi graph $\mathbb {G}(N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb {G}(N,r)$. For…

Almost all graphs with average degree 4 are 3-colorable

- Mathematics, Computer ScienceSTOC '02
- 2002

This work considers the problem of 3-coloring sparse random graphs and analyzes a "smoothed" version of the Brelaz heuristic to prove that almost all graphs with average degree d, i.e. G(n,p=d/n), are 3-colorable for d/n.

The Condensation Phase Transition in Random Graph Coloring

- Physics, Mathematics
- 2016

Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of…

Maximum independent sets on random regular graphs

- Mathematics, Physics
- 2013

We determine the asymptotics of the independence number of the random d-regular graph for all $${d\geq d_0}$$d≥d0. It is highly concentrated, with constant-order fluctuations around…

Sharp concentration of the chromatic number on random graphsGn, p

- Computer Science, MathematicsComb.
- 1987

The distribution of the chromatic number on random graphsGn, p is quite sharply concentrated. For fixedp it concentrates almost surely in √n ω(n) consecutive integers where ω(n) approaches infinity…

The condensation phase transition in random graph coloring

- Computer Science, MathematicsArXiv
- 2014

In random graph k-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem, and this paper proves this conjecture for k exceeding a certain constant k0.