Labelling planar graphs without 4-cycles with a condition on distance two

Let pp and qq be positive integers. An L(p,q)L(p,q)-labelling of a graph GG is a function ϕ:V(G)→N such that |ϕ(x)−ϕ(y)|⩾p|ϕ(x)−ϕ(y)|⩾p if xx and yy are adjacent and |ϕ(x)−ϕ(y)|⩾q|ϕ(x)−ϕ(y)|⩾q if xx and yy are of distance two apart. The L(p,q)L(p,q)-labelling number λ(G;p,q)λ(G;p,q) of GG is the least kk for which GG has an L(p,q)L(p,q)-labelling ϕ:V(G)→{0,1,…,k}ϕ:V(G)→{0,1,…,k}. In this paper we prove that for every planar graph GG without 4-cycles and of maximum degree Δ,λ(G;p,q)⩽min{(8q−4)Δ+8p−6q−1,(2q−1)Δ+10p+84q−47}Δ,λ(G;p,q)⩽min{(8q−4)Δ+8p−6q−1,(2q−1)Δ+10p+84q−47}.