The L(p,q)-labelling of planar graphs without 4-cycles

Wegner conjectured that for each planar graph G with maximum degree Δ at least 4, χ(G2)≤Δ+5 if 4≤Δ≤7, and χ(G2)≤⌊3Δ2⌋+1 if Δ≥8. Let G be a planar graph without 4-cycles. In this paper, we discuss the L(p,q)-labelling of G, and show that λp,q(G)≤(2q−1)Δ+8p+14q−11, where p and q are positive integers with p≥q. As a corollary, χ(G2)≤Δ+12 and λ2,1(G)≤Δ+19.