  We propose inhomogeneous random K-out graphs $\mathbb{H}(n; \pmb{\mu},
\pmb{K}_n)$, where each of the $n$ nodes is assigned to one of $r$ classes
independently with a probability distribution $\pmb{\mu} = \{\mu_1, \ldots,
\mu_r\}$. In particular, each node is classified as class-$i$ with probability
$\mu_i>0$, independently. Each class-$i$ node selects $K_{i,n}$ distinct nodes
uniformly at random from among all other nodes. A pair of nodes are adjacent in
$\mathbb{H}(n; \pmb{\mu}, \pmb{K}_n)$ if at least one selects the other.
Without loss of generality, we assume that $K_{1,n} \leq K_{2,n} \leq \ldots
\leq K_{r,n}$. Earlier results on homogeneous random K-out graphs
$\mathbb{H}(n; K_n)$, where all nodes select the same number $K$ of other
nodes, reveal that $\mathbb{H}(n; K_n)$ is connected with high probability
(whp) if $K_n \geq 2$ which implies that $\mathbb{H}(n; \pmb{\mu}, \pmb{K}_n)$
is connected whp if $K_{1,n} \geq 2$. In this paper, we investigate the
connectivity of inhomogeneous random K-out graphs $\mathbb{H}(n; \pmb{\mu},
\pmb{K}_n)$ for the special case when $K_{1,n}=1$, i.e., when each class-$1$
node selects only one other node. We show that
$\mathbb{H}\left(n;\pmb{\mu},\pmb{K}_n\right)$ is connected whp if $K_{r,n}$ is
chosen such that $\lim_{n \to \infty} K_{r,n} = \infty$. However, any bounded
choice of the sequence $K_{r,n}$ gives a positive probability of
$\mathbb{H}\left(n;\pmb{\mu},\pmb{K}_n\right)$ being not connected. Simulation
results are provided to validate our results in the finite node regime.
