Radius, Girth and Minimum Degree

TL;DR

This paper investigates the maximum radius of connected graphs with given minimum degree and girth, providing exact results for certain cases and establishing bounds and conjectural links for others.

Contribution

It determines the exact maximum radius for triangle-free graphs and establishes bounds and equivalences related to girth and radius for higher girth values.

Findings

Exact maximum radius for triangle-free graphs ($g=4$).

Order of maximum radius determined for $g=6,8,12$.

Upper bounds and conjectural links for general even girth.

Abstract

Given a connected graph $G$ on $n$ vertices, with minimum degree $\delta\geq 2$ and girth at least $g \geq 4$, what is the maximum radius $r$ this graph can have? Erd\H{o}s, Pach, Pollack and Tuza established in the triangle-free case ($g=4$) that $r \leq \frac{n-2}{\delta}+12$, and noted that up to the value of the additive constant, this is tight. We determine the exact value for the triangle-free case. For higher $g$ little is known. We settle the order of $r$ for $g=6,8,12$ and prove an upper bound to the order for general even $g$. Finally, we show that proving the corresponding lower bound for general even $g$ is equivalent to the Erd\H{o}s girth conjecture.