Minors in lifts of graphs

We study here lifts and random lifts of graphs, as defined by Amit and Linial (Combinatorica 22 (2002), 1-18). We consider the Hadwiger number η and the Hajós number σ of ℓ-lifts of Kⁿ and analyze their extremal as well as their typical values (that is, for random lifts). When ℓ = 2, we show that n/2 ≤ η ≤ n, and random lifts achieve the lower bound (as n → ∞). For bigger values of ℓ, we show Ω(n/√log n) ≤ η ≤ n √ℓ. We do not know how tight these bounds are, and in fact, the most interesting question that remains open is whether it is possible for η to be o(n). When ℓ ≤ O(log n), almost every ℓ-lift of Kⁿ satisfies η = ⊖(n) and for Ω(log n) ≤ ℓ ≤ ≤ n^1/3-ε, almost surely η = ⊖(n√ℓ/√log n). For bigger values of ℓ, Ω(n√ℓ/√log ℓ) ≤ η ≤ n√ℓ almost always. The Hajós number satisfies Ω(√n) ≤ σ ≤ n, and random lifts achieve the lower bound for bounded ℓ and approach the upper bound when ℓ grows.