k-regular subgraphs near the k-core threshold of a random graph

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 :is the random subgraph of G n 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 (r − 1)-degenerate, which is one fundamental graph property that has been shown widely applicable to various problems in graph theory. 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 (r − 1)-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.

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k-regular subgraphs near the k-core threshold of a random graph

Cited by 2 publications

References 22 publications

Threshold functions for small subgraphs in simple graphs and multigraphs

Threshold functions for small subgraphs in simple graphs and multigraphs

Phase Transition of Degeneracy in Minor-Closed Families

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