TY - JOUR

T1 - Random Lifts of Graphs

T2 - Independence and Chromatic Number

AU - Amit, Alon

AU - Linial, Nathan

AU - Matoušek, Jiří

PY - 2002/1

Y1 - 2002/1

N2 - 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 χf (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.

AB - 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 χf (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.

UR - http://www.scopus.com/inward/record.url?scp=0036103291&partnerID=8YFLogxK

U2 - 10.1002/rsa.10003

DO - 10.1002/rsa.10003

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AN - SCOPUS:0036103291

SN - 1042-9832

VL - 20

SP - 1

EP - 22

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 1

ER -