Abstract
. The (2,1)-total labelling of the tree has been widely studied. In this paper we study the (3,1)-total labelling number of the tree. The (3,1)-total labelling number of the tree is the width of the smallest range of integers to label the vertices and the edges such that no two adjacent vertices or two adjacent edges have the same labels and the difference between the labels of a vertex and its incident edges is at least 3. We prove that if the distance of the maximum degree in the tree is not 2, then the (3,1)-total labelling number is the maximum degree plus 3.
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