A mapping \(f\) of the vertices of graph \(G\) to the vertices of graph \(H\) is a homomorphism if \((f(u),f(v))\) is an edge of \(H\) for every edge \((u,v)\) of \(G\). This paper studies partial covers: these are homomorphisms that are `locally injective', and generalize the notion of graph cover. Amongst others, it is shown that for every \(k\)-regular graph \(H\) (\(k\geq 3\)), the problem to determine (given \(G\)) if a partial \(H\)-cover exists, is NP-complete. The notion is also related to the following problem: a \(H_{2,1}\) labeling \(f\) of \(G\) is a mapping of the vertices of \(G\) to \(H\) such that the distance in \(H\) between \(f(u)\) and \(f(v)\) is at least two, for adjacent \(u\) and \(v\) of \(G\), and vertices \(u\) and \(v\) with distance two in \(G\) have \(f(u)\) and \(f(v)\) with distance at least one in \(H\). This problem is a natural generalization of the well-studied lambda-coloring (also called radio coloring) problem.