Random Lifts of Graphs: Independence and Chromatic Number

For a graph G, a random n-lift of G has the vertex set V(G) × [n] and for each edge [u,v] ∈ E(G), there is a random matching between {u} × [n] and {v} × [n]. We present bounds on the chromatic number and on the independence number of typical random lifts, with G fixed and n → ∞ For the independence number, upper and lower bounds are obtained as solutions to certain optimization problems on the base graph. For a base graph G with chromatic number χ and fractional chromatic number χ _f, we show that the chromatic number of typical lifts is bounded from below by const ̇ √ χ/log χ and also by const ̇ χ/_f/log² χ_f (trivially, it is bounded by x from above). We have examples of graphs where the chromatic number of the lift equals x almost surely, and others where it is a.s. O(χ/log χ). Many interesting problems remain open.