The -labelling problem for trees

Let h ≥ 1 be an integer. An L ( h , 1 , 1 ) -labelling of a (finite or infinite) graph is an assignment of nonnegative integers (labels) to its vertices such that adjacent vertices receive labels with difference at least h , and vertices distance 2 or 3 apart receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over all L ( h , 1 , 1 ) -labellings is called the λ h , 1 , 1 -number of the graph. We prove that, for any integer h ≥ 1 and any finite tree T of diameter at least 3 or infinite tree T of finite maximum degree, max { max u v ∈ E ( T ) min { d ( u ) , d ( v ) } + h − 1 , Δ 2 ( T ) − 1 } ≤ λ h , 1 , 1 ( T ) ≤ Δ 2 ( T ) + h − 1 , and both lower and upper bounds are attainable, where Δ 2 ( T ) is the maximum total degree of two adjacent vertices. Moreover, if h is less than or equal to the minimum degree of a non-pendant vertex of T , then λ h , 1 , 1 ( T ) ≤ Δ 2 ( T ) + h − 2 . In particular, Δ 2 ( T ) − 1 ≤ λ 2 , 1 , 1 ( T ) ≤ Δ 2 ( T ) . Furthermore, if T is a caterpillar and h ≥ 2 , then max { max u v ∈ E ( T ) min { d ( u ) , d ( v ) } + h − 1 , Δ 2 ( T ) − 1 } ≤ λ h , 1 , 1 ( T ) ≤ Δ 2 ( T ) + h − 2 with both lower and upper bounds achievable.