image-total labelling of planar graphs with large girth and high maximum degree Original Research Article

The image-total number image 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, image-total labelling of graphs, in: Workshop on Graphs and Algorithms, Dijon, France, 2003]. In this paper, we prove that image for planar graphs with large girth and high maximum degree image. Our results are optimal for image.