Abstract
The ( d , 1 ) -total number λ d T ( 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 ) -total labelling of graphs, in: Workshop on Graphs and Algorithms, Dijon, France, 2003]. In this paper, we prove that λ d T ( G ) ⩽ Δ + 2 d - 2 for planar graphs with large girth and high maximum degree Δ . Our results are optimal for d = 2 .
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