Sudden emergence of q-regular subgraphs in random graphs

TL;DR

This paper studies the emergence of large q-regular subgraphs in random graphs using statistical physics methods, revealing a discontinuous transition near the 3-core percolation point.

Contribution

It introduces a novel approach to analyze q-regular subgraph emergence in random graphs via the cavity method, identifying critical thresholds and transition behaviors.

Findings

Large q-regular subgraphs appear discontinuously at specific average degrees.

The emergence threshold for q=3 is approximately 3.3546, close to the 3-core percolation point.

For q>3, the percolation threshold matches that of the q-core.

Abstract

We investigate the computationally hard problem whether a random graph of finite average vertex degree has an extensively large $q$-regular subgraph, i.e., a subgraph with all vertices having degree equal to $q$. We reformulate this problem as a constraint-satisfaction problem, and solve it using the cavity method of statistical physics at zero temperature. For $q=3$, we find that the first large $q$-regular subgraphs appear discontinuously at an average vertex degree $c_\reg{3} \simeq 3.3546$ and contain immediately about 24% of all vertices in the graph. This transition is extremely close to (but different from) the well-known 3-core percolation point $c_\cor{3} \simeq 3.3509$. For $q>3$, the $q$-regular subgraph percolation threshold is found to coincide with that of the $q$-core.