The image-labelling of trees

An image-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices have numbers at least 2 apart, and vertices at distance 2 have distinct numbers. The image-labelling number image of G is the minimum range of labels over all such labellings. It was shown by Griggs and Yeh [Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5 (1992) 586–595] that every tree T has image. This paper provides a sufficient condition for image. Namely, we prove that if a tree T contains no two vertices of maximum degree at distance either 1, 2, or 4, then image. Examples of trees T with two vertices of maximum degree at distance 4 such that image are constructed.