An nn-  L(δ1,δ2,δ3)L(δ1,δ2,δ3)labeling   of a simple graph G=(V,E)G=(V,E) is a mapping f:V→{0,1,…,n}f:V→{0,1,…,n} such that ∣f(u)−f(v)∣≥δi∣f(u)−f(v)∣≥δi when the distance between uu and vv is ii for i=1,2,3i=1,2,3. The L(δ1,δ2,δ3)L(δ1,δ2,δ3)labeling span  λ(δ1,δ2,δ3)(G)λ(δ1,δ2,δ3)(G) of a graph GG is the minimum nn such that GG admits an nn-L(δ1,δ2,δ3)L(δ1,δ2,δ3) labeling. In this article, we prove a conjecture by Calamoneri (2013) by showing λ(3,2,1)(L6)=19λ(3,2,1)(L6)=19 where L6L6 is the infinite triangular lattice. We also show that λ(4,2,1)(L6)=19λ(4,2,1)(L6)=19 but λ(k,2,1)(L6)≥20λ(k,2,1)(L6)≥20 for all k≥5k≥5.
['Sandip Das', 'Sasthi C. Ghosh', 'Soumen Nandi']
Optimal L(3,2,1)-labeling of triangular lattice