A note on cycles in graphs with specified radius and diameter

TL;DR

This paper provides a new proof that graphs with given radius and diameter contain cycles of a certain minimum length, specifically establishing a lower bound on the circumference based on these parameters.

Contribution

The paper introduces a novel proof technique to establish a lower bound on the circumference of graphs with specified radius and diameter.

Findings

Graphs with radius r and diameter d ≤ 2r-2 have a cycle of length at least 4r-2d.

The circumference c(G) is at least 4r-2d.

The proof offers a new perspective on cycle length bounds in graphs.

Abstract

Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.