Optimal L(3,2,1)-labeling of triangular lattice

An n-L(δ1,δ2,δ3) labeling of a simple graph G=(V,E) is a mapping f:V→{0,1,…,n} such that ∣f(u)−f(v)∣≥δi when the distance between u and v is i for i=1,2,3. The L(δ1,δ2,δ3) labeling span λ(δ1,δ2,δ3)(G) of a graph G is the minimum n such that G admits an n-L(δ1,δ2,δ3) labeling. In this article, we prove a conjecture by Calamoneri (2013) by showing λ(3,2,1)(L6)=19 where L6 is the infinite triangular lattice. We also show that λ(4,2,1)(L6)=19 but λ(k,2,1)(L6)≥20 for all k≥5.