Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3

TL;DR

This paper proves that all graphs with diameter 2 and minimum degree at least 3 contain cycles of length 4 or 8, supporting the Erdős-Gyárfás Conjecture for this class.

Contribution

It establishes a new result confirming the Erdős-Gyárfás Conjecture for diameter-2 graphs with minimum degree at least 3.

Findings

Graphs with diameter 2 and minimum degree ≥ 3 contain cycles of length 4 or 8

Supports the Erdős-Gyárfás Conjecture for this graph class

Provides a structural insight into cycle lengths in specific graphs

Abstract

In this short note it is shown that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8. This result contributes to the study of the Erd\H{o}s-Gy\'arf\'as Conjecture by confirming it for the class of diameter-2 graphs.