Random Lifts of Graphs III: Independence and Chromatic Number
نویسندگان
چکیده
For a graph G, a random n-lift of G has the vertex set V (G) n] and for each edge u; v] 2 E(G), there is a random matching between fug n] and fvg n]. We present bounds on the chromatic number and on the independence number of typical random lifts, with G xed and n ! 1. 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 and fractional chromatic number f , we show that the chromatic number of typical lifts is bounded from below by const p = log and also by const f = log 2 f (trivially, it is bounded by from above). We have examples of graphs where the chromatic number of the lift equals almost surely, and others where it is a.s. O(= log). Many interesting problems remain open.
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