Every graph contains a linearly sized induced subgraph with all degrees odd

TL;DR

This paper proves that every sufficiently large graph without isolated vertices contains a linearly sized induced subgraph where all vertices have odd degrees, resolving a longstanding conjecture in graph theory.

Contribution

It establishes that any graph with no isolated vertices has a large induced subgraph with all degrees odd, confirming a well-known conjecture.

Findings

Every graph with no isolated vertices has an induced subgraph of size at least n/10000 with all degrees odd.

The result applies to all graphs without isolated vertices, regardless of their structure.

This solves a long-standing open problem in graph theory.

Abstract

We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.