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A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length. While the tetravalent edge-transitive graphs admitting a semiregular automorphism with only one orbit are easy to determine, those that admit a semiregular automorphism with two orbits took a considerable effort and were finally classified in 2012. Of the several possible different ``types'' of potential tetravalent edge-transitive graphs admitting a semiregular automorphism with three orbits, only one ``type'' has thus far received no attention. In this paper we focus on this class of graphs, which we call the Woolly Hat graphs. We prove that there are in fact no edge-transitive Woolly Hat graphs and classify the vertex-transitive ones., Comment: 26 pages

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph $\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $\bcay(G,T),$ $\bcay(G,S) \cong \bcay(G,T)$ implies that $T = g S^\alpha$ for some $g \in G$ and $\alpha \in \aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups $\Z_{2^r},$ $\Z_2^r$ and $\Q_8$.