Erdős–Gyárfás Conjecture: Power-of-two cycles in graphs with minimum degree at least 3

Prove that every graph with minimum degree at least 3 contains a cycle whose length is a power of 2.

Background

The paper establishes that every graph of diameter 2 with minimum degree at least 3 contains a 4-cycle or an 8-cycle. This verifies the Erdős–Gyárfás Conjecture for the subclass of graphs of diameter 2.

The Erdős–Gyárfás Conjecture is a longstanding question in graph theory asserting the existence of a cycle of length a power of two in any graph with minimum degree at least 3. The authors explicitly restate this conjecture and position their main theorem as partial progress—confirming the conjecture within a specific structural regime (diameter 2).