(d,1)(d,1)-total labelling of planar graphs with large girth and high maximum degree

The (d,1)(d,1)-total number λdT(G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least d  . This notion was introduced in [F. Havet, (d,1)(d,1)-total labelling of graphs, in: Workshop on Graphs and Algorithms, Dijon, France, 2003]. In this paper, we prove that λdT(G)⩽Δ+2d-2 for planar graphs with large girth and high maximum degree ΔΔ. Our results are optimal for d=2d=2.