  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 $\Q_8$.

Group Theory;Combinatorics