  Minimal multicast networks are fascinating and efficient combinatorial
objects, where the removal of a single link makes it impossible for all
receivers to obtain all messages. We study the structure of such networks, and
prove some constraints on their possible solutions.
  We then focus on the combination network, which is one of the simplest and
most insightful network in network-coding theory. Of particular interest are
minimal combination networks. We study the gap in alphabet size between
vector-linear and scalar-linear network-coding solutions for such minimal
combination networks and some of their sub-networks.
  For minimal multicast networks with two source messages we find the maximum
possible gap. We define and study sub-networks of the combination network,
which we call Kneser networks, and prove that they attain the upper bound on
the gap with equality. We also prove that the study of this gap may be limited
to the study of sub-networks of minimal combination networks, by using graph
homomorphisms connected with the $q$-analog of Kneser graphs. Additionally, we
prove a gap for minimal multicast networks with three or more source messages
by studying Kneser networks.
  Finally, an upper bound on the gap for full minimal combination networks
shows nearly no gap, or none in some cases. This is obtained using an MDS-like
bound for subspaces over a finite field.
