Given an infinite family G of graphs and a monotone property
P, an (upper) threshold for G and P is a
"fastest growing" function p:Nβ[0,1] such that limnβββPr(Gnβ(p(n))βP)=1 for any sequence (Gnβ)nβNβ over G with limnββββ£V(Gnβ)β£=β, where Gnβ(p(n)) is the random subgraph of Gnβ such that each
edge remains independently with probability p(n).
In this paper we study the upper threshold for the family of H-minor free
graphs and for the graph property of being (rβ1)-degenerate, which is one
fundamental graph property with many applications. Even a constant factor
approximation for the upper threshold for all pairs (r,H) is expected to be
very difficult by its close connection to a major open question in extremal
graph theory. We determine asymptotically the thresholds (up to a constant
factor) for being (rβ1)-degenerate for a large class of pairs (r,H),
including all graphs H of minimum degree at least r and all graphs H with
no vertex-cover of size at most r, and provide lower bounds for the rest of
the pairs of (r,H). The results generalize to arbitrary proper minor-closed
families and the properties of being r-colorable, being r-choosable, or
containing an r-regular subgraph, respectively