On subgraphs with degrees of prescribed residues in the random graph

TL;DR

This paper proves that in a random graph, large induced subgraphs with degrees of prescribed residues modulo q exist with high probability, and also shows a partition into such subgraphs, confirming a conjecture for q=2.

Contribution

It establishes the existence of large induced subgraphs with degrees of specified residues modulo q and a partition into such subgraphs, generalizing previous conjectures.

Findings

Existence of large induced subgraphs with degrees congruent to r mod q

Partition of the graph into q+1 parts with degrees congruent to r mod q

Resolution of Scott's conjecture for q=2

Abstract

We show that with high probability the random graph $G_{n, 1/2}$ has an induced subgraph of linear size, all of whose degrees are congruent to $r\pmod q$ for any fixed $r$ and $q\geq 2$. More generally, the same is true for any fixed distribution of degrees modulo $q$. Finally, we show that with high probability we can partition the vertices of $G_{n, 1/2}$ into $q+1$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to $r\pmod q$. Our results resolve affirmatively a conjecture of Scott, who addressed the case $q=2$.