Distance constrained labelings of planar graphs with no short cycles

Motivated by a conjecture of Wang and Lih, we show that every planar graph of girth at least seven and maximum degree Δ≥190+2⌈p/q⌉Δ≥190+2⌈p/q⌉ has an L(p,q)L(p,q)-labeling of span at most 2p+qΔ−22p+qΔ−2. Since the optimal span of an L(p,1)L(p,1)-labeling of an infinite ΔΔ-regular tree is 2p+Δ−22p+Δ−2, the obtained bound is the best possible for any p≥1p≥1 and q=1q=1.