A new sufficient condition for a tree T to have the (2,1)-total number

A k-(2, 1)-total labelling of a graph G is a mapping such that adjacent vertices or adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least 2. The (2, 1)-total number, denoted , is the minimum k such that G has a k-(2, 1)-total labelling. Let T be a tree with maximum degree . A vertex is called major if , minor if , and saturated if v is major and is adjacent to exactly major vertices. It is known that . In this paper, we prove that if every major vertex is adjacent to at most major vertices, and every minor vertex is adjacent to at most three saturated vertices, then . The result is best possible with respect to these required conditions.