A complete L(2,1)L(2,1) span characterization for small trees

An L(2,1)L(2,1) labeling of a graph GG is a vertex labeling such that any pair of vertices vivi and vjvj must have labels at least 2 apart if d(vi,vj)=1d(vi,vj)=1 and labels at least 1 apart if d(vi,vj)=2d(vi,vj)=2. The span of an L(2,1)L(2,1) labeling ff on a graph GG is the maximum f(u)f(u) for all u∈V(G)u∈V(G). The L(2,1)L(2,1) span of a graph GG is the minimum span of all L(2,1)L(2,1) labelings on GG. The L(2,1)L(2,1) labeling on trees has been extensively studied in recent years. In this paper we present a complete characterization of the L(2,1)L(2,1) span of trees up to twenty vertices.