A Formula: see text-Formula: see text-labelling of graph Formula: see text is a function Formula: see text, and satisfy the following two requirements (1) if Formula: see text, then Formula: see text; (2) if Formula: see text and Formula: see text are of distance two apart, then Formula: see text, where Formula: see text and Formula: see text are integers with Formula: see text. There has been extensive research on Formula: see text-labelling of graphs. Let Formula: see text be a planar graph with maximum degree Formula: see text, and Formula: see text. We prove that Formula: see text, which improves the bound given by Dong3.
Jin et al. (Fri,) studied this question.