Facial L(2,1)-edge-labelings of trees

In this paper, we prove that the facial L(2,1)-edge-labeling index of any tree T is at most 7; moreover, this bound is tight. In the case when T has no vertex of degree 3 the upper bound for this parameter is 6, which is also tight. If T is without vertices of degree 2 and 3, then its facial L(2,1)-edge-labeling index is at most 5; moreover, this bound is also tight. Finally, we characterize all trees having facial L(2,1)-edge-labeling index exactly 4.