For a graph G, an L(2,1)-labeling of G with span k is a mapping L \right arrow \{0, 1, 2, \ldots, k\} such that adjacent vertices are assigned integers which differ by at least 2, vertices at distance two are assigned integers which differ by at least 1, and the image of L includes 0 and k. The minimum span over all L(2,1)-labelings of G is denoted λ(G), and each L(2,1)-labeling with span λ(G) is called a λ-labeling. For h∈{1,…,k−1}, h is a hole of Lif and only if h is not in the image of L. The minimum number of holes over all λ-labelings is denoted ρ(G), and the minimum k for which there exists a surjective L(2,1)-labeling onto {0,1, ..., k} is denoted μ(G). This paper extends the work of Fishburn and Roberts on ρ and μ through the investigation of an equivalence relation on the set of λ-labelings with ρ holes. In particular, we establish that ρ≤Δ. We analyze the structure of those graphs for which ρ∈{Δ−1,Δ}, and we show that μ=λ+1 whenever λ is less than the order of the graph. Finally, we give constructions of connected graphs with ρ=Δ and order t(Δ+1), 1≤t≤Δ