4 research outputs found
k-regular subgraphs near the k-core threshold of a random graph
We prove that whp has a -regular subgraph if is at least
above the threshold for the appearance of a subgraph with
minimum degree at least ; i.e. an non-empty -core. In particular, this
pins down the threshold for the appearance of a -regular subgraph to a
window of size
Phase Transition of Degeneracy in Minor-Closed Families
Given an infinite family of graphs and a monotone property
, an (upper) threshold for and is a
"fastest growing" function such that for any sequence over with , where is the random subgraph of such that each
edge remains independently with probability .
In this paper we study the upper threshold for the family of -minor free
graphs and for the graph property of being -degenerate, which is one
fundamental graph property with many applications. Even a constant factor
approximation for the upper threshold for all pairs 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 -degenerate for a large class of pairs ,
including all graphs of minimum degree at least and all graphs with
no vertex-cover of size at most , and provide lower bounds for the rest of
the pairs of . The results generalize to arbitrary proper minor-closed
families and the properties of being -colorable, being -choosable, or
containing an -regular subgraph, respectively