Distance constrained labelings of -minor free graphs

Motivated by previous results on distance constrained labelings and coloring of squares of K 4 -minor free graphs, we show that for every p ≥ q ≥ 1 , there exists Δ 0 such that every K 4 -minor free graph G with maximum degree Δ ≥ Δ 0 has an L ( p , q ) -labeling of span at most q ⌊ 3 Δ ( G ) / 2 ⌋ . The obtained bound is the best possible.