# On the radius and the attachment number of tetravalent half-arc-transitive graphs

## TL;DR

This paper investigates the relationship between radius and attachment number in tetravalent half-arc-transitive graphs, proving specific divisibility properties and characterizing certain graph classes.

## Contribution

It establishes that if the attachment number is twice an odd number, then it divides twice the radius, and characterizes graphs with radius 3 and attachment number 2 as certain covers of line graphs.

## Key findings

- If a is twice an odd number, then a divides 2r.
- Graphs with r=3 and a=2 are non-sectional split 2-fold covers of line graphs of 2-arc-transitive cubic graphs.
- Confirmed that all graphs with a not dividing r are arc-transitive in the studied cases.

## Abstract

In this paper, we study the relationship between the radius $r$ and the attachment number $a$ of a tetravalent graph admitting a half-arc-transitive group of automorphisms. These two parameters were first introduced in~[{\em J.~Combin.~Theory Ser.~B} {73} (1998), 41--76], where among other things it was proved that $a$ always divides $2r$. Intrigued by the empirical data from the census~[{\em Ars Math.\ Contemp.} {8} (2015)] of all such graphs of order up to 1000 we pose the question of whether all examples for which $a$ does not divide $r$ are arc-transitive. We prove that the answer to this question is positive in the case when $a$ is twice an odd number. In addition, we completely characterize the tetravalent graphs admitting a half-arc-transitive group with $r = 3$ and $a=2$, and prove that they arise as non-sectional split $2$-fold covers of line graphs of $2$-arc-transitive cubic graphs.