The L(h, 1, 1)-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 λh,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{maxuv∈E(T )min{d(u), d(v)} + h − 1,∆2(T ) − 1} ≤ λh,1,1(T ) ≤ ∆2(T ) + h − 1, and both lower and upper bounds are attainable, where∆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 λh,1,1(T ) ≤ ∆2(T ) + h − 2. In particular, ∆2(T ) − 1 ≤ λ2,1,1(T ) ≤ ∆2(T ). Furthermore, if T is a caterpillar and h ≥ 2, thenmax{maxuv∈E(T )min{d(u), d(v)} + h− 1,∆2(T )− 1} ≤ λh,1,1(T ) ≤ ∆2(T )+ h− 2 with both lower and upper bounds achievable. © 2009 Elsevier Ltd. All rights reserved.