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arXiv · 2009.00741

Radius, Girth and Minimum Degree

Vojtěch Dvořák·Peter van Hintum·Amy Shaw·Marius Tiba
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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.

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Vojtěch Dvořák, Peter van Hintum, Amy Shaw, Marius Tiba. 2020-09-01. Radius, Girth and Minimum Degree. https://arxiv.org/abs/2009.00741

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