# Chromatic number of random lifts of complete graphs

**Question**Is the chromatic number of a random lift of concentrated on a single value?

Let be a graph with vertex set and edge set . An -lift is a graph with vertex set , such that and may only be adjacent in if , and for each , the edges between and form a perfect matching.

A random -lift of is a graph drawn uniformly at random from the set of all -lifts of . This amounts to choosing, independently at random, a perfect matching for each edge of . One is generally interested in properties of random -lifts when .

Amit, Linial, and Matousek [ALM02] have studied the chromatic number of random lifts. They ask whether a the chromatic number of a random -lift of is asymptotically almost surely a single number.

It is easy to see that this number may be either 3 or 4. Farzad and Theis [FT12] have shown that random lifts of are asymptotically almost surely 3-colorable.

A more general question is this.

**Question**Is the chromatic number of a random lift of concentrated on a single value?

Amit, Linial, and Matousek [ALM02] have shown that the chromatic number of a random lift of is in .

## Bibliography

*[ALM02] Random Lifts of Graphs III: Independence and Chromatic Number, A. Amit, N. Linial and J. Matousek, Random Structures and Algorithms, 20(2002) 1-22.

[FT12] Random lifts of are 3-colourable. B. Farzad and D.O. Theis. SIAM J. Discrete Math. 26:1 (2012), 169–179.

* indicates original appearance(s) of problem.