Phase Transition of Degeneracy in Minor-Closed Families

TL;DR

This paper investigates the phase transition thresholds for the property of being (r-1)-degenerate in random subgraphs of minor-closed graph families, providing asymptotic results for various classes of graphs.

Contribution

It determines asymptotic thresholds for (r-1)-degeneracy in minor-closed families for broad classes of graphs, extending to properties like colorability and containing r-regular subgraphs.

Findings

Thresholds are asymptotically determined for graphs with minimum degree at least r.

Results include all graphs with no small vertex cover.

Lower bounds are provided for remaining pairs (r, H).

Abstract

Given an infinite family ${\mathcal G}$ of graphs and a monotone property ${\mathcal P}$, an (upper) threshold for ${\mathcal G}$ and ${\mathcal P}$ is a "fastest growing" function $p: \mathbb{N} \to [0,1]$ such that $\lim_{n \to \infty} \Pr(G_n(p(n)) \in {\mathcal P})= 1$ for any sequence $(G_n)_{n \in \mathbb{N}}$ over ${\mathcal G}$ with $\lim_{n \to \infty}\lvert V(G_n) \rvert = \infty$, where $G_n(p(n))$ 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 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 $(r-1)$-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.